diff --git a/工具/关键字筛选题号.ipynb b/工具/关键字筛选题号.ipynb index fb447156..d8a48d79 100644 --- a/工具/关键字筛选题号.ipynb +++ b/工具/关键字筛选题号.ipynb @@ -2,7 +2,7 @@ "cells": [ { "cell_type": "code", - "execution_count": 3, + "execution_count": 11, "metadata": {}, "outputs": [ { @@ -11,7 +11,7 @@ "0" ] }, - "execution_count": 3, + "execution_count": 11, "metadata": {}, "output_type": "execute_result" } @@ -21,7 +21,7 @@ "\n", "\"\"\"---设置关键字, 同一field下不同选项为or关系, 同一字典中不同字段间为and关系, 不同字典间为or关系, _not表示列表中的关键字都不含, 同一字典中的数字用来供应同一字段不同的条件之间的and---\"\"\"\n", "keywords_dict_table = [\n", - " {\"tags\":[\"空间向量\"]}\n", + " {\"tags\":[\"第七单元\"],\"tags_not\":[\"抛物线\",\"椭圆\",\"双曲线\",\"直线\"],\"content_not\":[\"圆\"]}\n", "]\n", "\"\"\"---关键字设置完毕---\"\"\"\n", "# 示例： keywords_dict_table = [\n", diff --git a/工具/分年级专用工具/赋能卷生成.ipynb b/工具/分年级专用工具/赋能卷生成.ipynb index b960e6f9..2f76fa7b 100644 --- a/工具/分年级专用工具/赋能卷生成.ipynb +++ b/工具/分年级专用工具/赋能卷生成.ipynb @@ -2,7 +2,7 @@ "cells": [ { "cell_type": "code", - "execution_count": 2, + "execution_count": 10, "metadata": {}, "outputs": [ { @@ -17,11 +17,11 @@ "#在 临时文件/赋能答题纸 目录中保留一个pdf(赋能试卷的答题纸), 不留别的pdf文件. \n", "#在 临时文件/赋能答题纸 目录中保留 赋能template.tex.\n", "\"\"\"---设置文件名---\"\"\"\n", - "filename = \"赋能05\"\n", + "filename = \"赋能09\"\n", "\n", "\"\"\"---设置题目列表---\"\"\"\n", "problems = r\"\"\"\n", - "366:369,30281,371:375\n", + "406:408,11585,410:415\n", "\"\"\"\n", "#完成后将含有 filename 的文件移至其它目录\n", "\n", diff --git a/工具/寻找阶段末尾空闲题号.ipynb b/工具/寻找阶段末尾空闲题号.ipynb index 1f6a48e8..444ca2a0 100644 --- a/工具/寻找阶段末尾空闲题号.ipynb +++ b/工具/寻找阶段末尾空闲题号.ipynb @@ -2,7 +2,7 @@ "cells": [ { "cell_type": "code", - "execution_count": 3, + "execution_count": 4, "metadata": {}, "outputs": [ { @@ -11,7 +11,7 @@ "text": [ "首个空闲id: 12030 , 直至 020000\n", "首个空闲id: 20227 , 直至 030000\n", - "首个空闲id: 30473 , 直至 999999\n" + "首个空闲id: 30474 , 直至 999999\n" ] } ], diff --git a/工具/批量添加题库字段数据.ipynb b/工具/批量添加题库字段数据.ipynb index 40092aa2..c855d823 100644 --- a/工具/批量添加题库字段数据.ipynb +++ b/工具/批量添加题库字段数据.ipynb @@ -2,26 +2,161 @@ "cells": [ { "cell_type": "code", - "execution_count": 18, + "execution_count": 23, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ - "题号: 010540 , 字段: usages 中已添加数据: 20221105\t2023届高三02班\t0.839\t0.774\t0.710\n", - "题号: 010535 , 字段: usages 中已添加数据: 20221105\t2023届高三02班\t1.000\t1.000\t1.000\n", - "题号: 000222 , 字段: usages 中已添加数据: 20221105\t2023届高三02班\t0.935\n", - "题号: 000654 , 字段: usages 中已添加数据: 20221105\t2023届高三02班\t0.548\n", - "题号: 003585 , 字段: usages 中已添加数据: 20221105\t2023届高三02班\t0.774\n", - "题号: 003787 , 字段: usages 中已添加数据: 20221105\t2023届高三02班\t0.968\n", - "题号: 004647 , 字段: usages 中已添加数据: 20221105\t2023届高三02班\t0.968\n", - "题号: 009349 , 字段: usages 中已添加数据: 20221105\t2023届高三02班\t0.839\n", - "题号: 000229 , 字段: usages 中已添加数据: 20221105\t2023届高三02班\t0.903\n", - "题号: 000672 , 字段: usages 中已添加数据: 20221105\t2023届高三02班\t0.935\n", - "题号: 009741 , 字段: usages 中已添加数据: 20221105\t2023届高三02班\t0.903\n", - "题号: 010545 , 字段: usages 中已添加数据: 20221105\t2023届高三02班\t1.000\n", - "题号: 009744 , 字段: usages 中已添加数据: 20221105\t2023届高三02班\t0.710\n" + "题号: 000275 , 字段: tags 中已添加数据: 抛物线\n", + "题号: 000279 , 字段: tags 中已添加数据: 抛物线\n", + "题号: 000337 , 字段: tags 中已添加数据: 抛物线\n", + "题号: 000369 , 字段: tags 中已添加数据: 抛物线\n", + "题号: 000464 , 字段: tags 中已添加数据: 抛物线\n", + "题号: 000467 , 字段: tags 中已添加数据: 抛物线\n", + "题号: 000669 , 字段: tags 中已添加数据: 抛物线\n", + "题号: 000707 , 字段: tags 中已添加数据: 抛物线\n", + "题号: 000728 , 字段: tags 中已添加数据: 抛物线\n", + "题号: 000739 , 字段: tags 中已添加数据: 抛物线\n", + "题号: 000783 , 字段: tags 中已添加数据: 抛物线\n", + "题号: 000804 , 字段: tags 中已添加数据: 抛物线\n", + "题号: 000806 , 字段: tags 中已添加数据: 抛物线\n", + "题号: 000833 , 字段: tags 中已添加数据: 抛物线\n", + "题号: 000864 , 字段: tags 中已添加数据: 抛物线\n", + "题号: 000878 , 字段: tags 中已添加数据: 抛物线\n", + "题号: 000909 , 字段: tags 中已添加数据: 抛物线\n", + "题号: 000957 , 字段: tags 中已添加数据: 抛物线\n", + "题号: 000968 , 字段: tags 中已添加数据: 抛物线\n", + "题号: 002393 , 字段: tags 中已添加数据: 抛物线\n", + "题号: 002395 , 字段: tags 中已添加数据: 抛物线\n", + "题号: 002397 , 字段: tags 中已添加数据: 抛物线\n", + "题号: 002398 , 字段: tags 中已添加数据: 抛物线\n", + "题号: 002400 , 字段: tags 中已添加数据: 抛物线\n", + "题号: 002401 , 字段: tags 中已添加数据: 抛物线\n", + "题号: 002402 , 字段: tags 中已添加数据: 抛物线\n", + "题号: 002403 , 字段: tags 中已添加数据: 抛物线\n", + "题号: 002404 , 字段: tags 中已添加数据: 抛物线\n", + "题号: 002405 , 字段: tags 中已添加数据: 抛物线\n", + "题号: 002406 , 字段: tags 中已添加数据: 抛物线\n", + "题号: 002407 , 字段: tags 中已添加数据: 抛物线\n", + "题号: 002408 , 字段: tags 中已添加数据: 抛物线\n", + "题号: 002409 , 字段: tags 中已添加数据: 抛物线\n", + "题号: 002410 , 字段: tags 中已添加数据: 抛物线\n", + "题号: 002411 , 字段: tags 中已添加数据: 抛物线\n", + "题号: 002412 , 字段: tags 中已添加数据: 抛物线\n", + "题号: 002413 , 字段: tags 中已添加数据: 抛物线\n", + "题号: 002417 , 字段: tags 中已添加数据: 抛物线\n", + "题号: 002418 , 字段: tags 中已添加数据: 抛物线\n", + "题号: 002419 , 字段: tags 中已添加数据: 抛物线\n", + "题号: 002420 , 字段: tags 中已添加数据: 抛物线\n", + "题号: 002421 , 字段: tags 中已添加数据: 抛物线\n", + "题号: 002422 , 字段: tags 中已添加数据: 抛物线\n", + "题号: 002423 , 字段: tags 中已添加数据: 抛物线\n", + "题号: 002424 , 字段: tags 中已添加数据: 抛物线\n", + "题号: 002425 , 字段: tags 中已添加数据: 抛物线\n", + "题号: 002426 , 字段: tags 中已添加数据: 抛物线\n", + "题号: 002427 , 字段: tags 中已添加数据: 抛物线\n", + "题号: 002429 , 字段: tags 中已添加数据: 抛物线\n", + "题号: 002430 , 字段: tags 中已添加数据: 抛物线\n", + "题号: 002431 , 字段: tags 中已添加数据: 抛物线\n", + "题号: 002432 , 字段: tags 中已添加数据: 抛物线\n", + "题号: 002434 , 字段: tags 中已添加数据: 抛物线\n", + "题号: 002436 , 字段: tags 中已添加数据: 抛物线\n", + "题号: 002437 , 字段: tags 中已添加数据: 抛物线\n", + "题号: 002438 , 字段: tags 中已添加数据: 抛物线\n", + "题号: 002439 , 字段: tags 中已添加数据: 抛物线\n", + "题号: 002440 , 字段: tags 中已添加数据: 抛物线\n", + "题号: 002441 , 字段: tags 中已添加数据: 抛物线\n", + "题号: 002445 , 字段: tags 中已添加数据: 抛物线\n", + "题号: 002450 , 字段: tags 中已添加数据: 抛物线\n", + "题号: 002685 , 字段: tags 中已添加数据: 抛物线\n", + "题号: 002689 , 字段: tags 中已添加数据: 抛物线\n", + "题号: 002690 , 字段: tags 中已添加数据: 抛物线\n", + "题号: 003437 , 字段: tags 中已添加数据: 抛物线\n", + "题号: 003438 , 字段: tags 中已添加数据: 抛物线\n", + "题号: 003439 , 字段: tags 中已添加数据: 抛物线\n", + "题号: 003440 , 字段: tags 中已添加数据: 抛物线\n", + "题号: 003441 , 字段: tags 中已添加数据: 抛物线\n", + "题号: 003443 , 字段: tags 中已添加数据: 抛物线\n", + "题号: 003444 , 字段: tags 中已添加数据: 抛物线\n", + "题号: 003446 , 字段: tags 中已添加数据: 抛物线\n", + "题号: 003447 , 字段: tags 中已添加数据: 抛物线\n", + "题号: 003448 , 字段: tags 中已添加数据: 抛物线\n", + "题号: 003449 , 字段: tags 中已添加数据: 抛物线\n", + "题号: 003450 , 字段: tags 中已添加数据: 抛物线\n", + "题号: 003451 , 字段: tags 中已添加数据: 抛物线\n", + "题号: 003599 , 字段: tags 中已添加数据: 抛物线\n", + "题号: 003639 , 字段: tags 中已添加数据: 抛物线\n", + "题号: 003781 , 字段: tags 中已添加数据: 抛物线\n", + "题号: 003795 , 字段: tags 中已添加数据: 抛物线\n", + "题号: 003837 , 字段: tags 中已添加数据: 抛物线\n", + "题号: 003930 , 字段: tags 中已添加数据: 抛物线\n", + "题号: 003945 , 字段: tags 中已添加数据: 抛物线\n", + "题号: 004065 , 字段: tags 中已添加数据: 抛物线\n", + "题号: 004078 , 字段: tags 中已添加数据: 抛物线\n", + "题号: 004141 , 字段: tags 中已添加数据: 抛物线\n", + "题号: 004197 , 字段: tags 中已添加数据: 抛物线\n", + "题号: 004221 , 字段: tags 中已添加数据: 抛物线\n", + "题号: 004225 , 字段: tags 中已添加数据: 抛物线\n", + "题号: 004351 , 字段: tags 中已添加数据: 抛物线\n", + "题号: 004372 , 字段: tags 中已添加数据: 抛物线\n", + "题号: 004495 , 字段: tags 中已添加数据: 抛物线\n", + "题号: 004514 , 字段: tags 中已添加数据: 抛物线\n", + "题号: 004524 , 字段: tags 中已添加数据: 抛物线\n", + "题号: 004550 , 字段: tags 中已添加数据: 抛物线\n", + "题号: 004570 , 字段: tags 中已添加数据: 抛物线\n", + "题号: 004626 , 字段: tags 中已添加数据: 抛物线\n", + "题号: 004639 , 字段: tags 中已添加数据: 抛物线\n", + "题号: 004654 , 字段: tags 中已添加数据: 抛物线\n", + "题号: 004713 , 字段: tags 中已添加数据: 抛物线\n", + "题号: 004764 , 字段: tags 中已添加数据: 抛物线\n", + "题号: 008920 , 字段: tags 中已添加数据: 抛物线\n", + "题号: 008921 , 字段: tags 中已添加数据: 抛物线\n", + "题号: 008922 , 字段: tags 中已添加数据: 抛物线\n", + "题号: 008923 , 字段: tags 中已添加数据: 抛物线\n", + "题号: 008924 , 字段: tags 中已添加数据: 抛物线\n", + "题号: 008925 , 字段: tags 中已添加数据: 抛物线\n", + "题号: 008926 , 字段: tags 中已添加数据: 抛物线\n", + "题号: 008927 , 字段: tags 中已添加数据: 抛物线\n", + "题号: 008928 , 字段: tags 中已添加数据: 抛物线\n", + "题号: 008929 , 字段: tags 中已添加数据: 抛物线\n", + "题号: 008930 , 字段: tags 中已添加数据: 抛物线\n", + "题号: 008931 , 字段: tags 中已添加数据: 抛物线\n", + "题号: 008932 , 字段: tags 中已添加数据: 抛物线\n", + "题号: 008933 , 字段: tags 中已添加数据: 抛物线\n", + "题号: 008934 , 字段: tags 中已添加数据: 抛物线\n", + "题号: 008935 , 字段: tags 中已添加数据: 抛物线\n", + "题号: 008936 , 字段: tags 中已添加数据: 抛物线\n", + "题号: 008937 , 字段: tags 中已添加数据: 抛物线\n", + "题号: 008953 , 字段: tags 中已添加数据: 抛物线\n", + "题号: 008954 , 字段: tags 中已添加数据: 抛物线\n", + "题号: 008955 , 字段: tags 中已添加数据: 抛物线\n", + "题号: 008956 , 字段: tags 中已添加数据: 抛物线\n", + "题号: 008957 , 字段: tags 中已添加数据: 抛物线\n", + "题号: 008960 , 字段: tags 中已添加数据: 抛物线\n", + "题号: 008966 , 字段: tags 中已添加数据: 抛物线\n", + "题号: 008968 , 字段: tags 中已添加数据: 抛物线\n", + "题号: 009077 , 字段: tags 中已添加数据: 抛物线\n", + "题号: 009081 , 字段: tags 中已添加数据: 抛物线\n", + "题号: 009098 , 字段: tags 中已添加数据: 抛物线\n", + "题号: 009099 , 字段: tags 中已添加数据: 抛物线\n", + "题号: 009105 , 字段: tags 中已添加数据: 抛物线\n", + "题号: 009106 , 字段: tags 中已添加数据: 抛物线\n", + "题号: 009110 , 字段: tags 中已添加数据: 抛物线\n", + "题号: 009836 , 字段: tags 中已添加数据: 抛物线\n", + "题号: 009837 , 字段: tags 中已添加数据: 抛物线\n", + "题号: 009838 , 字段: tags 中已添加数据: 抛物线\n", + "题号: 009839 , 字段: tags 中已添加数据: 抛物线\n", + "题号: 009840 , 字段: tags 中已添加数据: 抛物线\n", + "题号: 010682 , 字段: tags 中已添加数据: 抛物线\n", + "题号: 010683 , 字段: tags 中已添加数据: 抛物线\n", + "题号: 010684 , 字段: tags 中已添加数据: 抛物线\n", + "题号: 010685 , 字段: tags 中已添加数据: 抛物线\n", + "题号: 010686 , 字段: tags 中已添加数据: 抛物线\n", + "题号: 010688 , 字段: tags 中已添加数据: 抛物线\n", + "题号: 010689 , 字段: tags 中已添加数据: 抛物线\n", + "题号: 010703 , 字段: tags 中已添加数据: 抛物线\n" ] } ], diff --git a/工具/批量题号选题pdf生成.ipynb b/工具/批量题号选题pdf生成.ipynb index bd7a7d55..fbe5247f 100644 --- a/工具/批量题号选题pdf生成.ipynb +++ b/工具/批量题号选题pdf生成.ipynb @@ -2,16 +2,16 @@ "cells": [ { "cell_type": "code", - "execution_count": 1, + "execution_count": 2, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ - "开始编译教师版本pdf文件: 临时文件/批量生成题目/立体几何1批量.tex_教师用_20221105.tex\n", + "开始编译教师版本pdf文件: 临时文件/赋能批量_教师用_20221105.tex\n", "0\n", - "开始编译学生版本pdf文件: 临时文件/批量生成题目/立体几何1批量.tex_学生用_20221105.tex\n", + "开始编译学生版本pdf文件: 临时文件/赋能批量_学生用_20221105.tex\n", "0\n" ] } @@ -26,19 +26,18 @@ "\"\"\"---设置题目列表---\"\"\"\n", "#字典字段为文件名, 之后为内容的题号\n", "problems_dict = {\n", - "\"K0629001X\":\"000293,000297,009868,010740,030465\",\n", - "\"K0630002X\":\"000298,000302,000304,001948,004348,009870,010721,010730\",\n", - "\"K0630004X\":\"000294,000300,003647,004656,004698,004740,009871,010732,010735,030462,030472\",\n", - "\"K0631002X\":\"000303,000305,001981,004243,009873,010739\",\n", - "\"K0631003X\":\"000295,000296,000299,004096,009872,010737,030461,030468\"\n", + "\"赋能6\":\"376:385\",\n", + "\"赋能7\":\"386:395\",\n", + "\"赋能8\":\"396:405\",\n", + "\"赋能9\":\"406:415\"\n", "}\n", "\n", "\"\"\"---设置题目列表结束---\"\"\"\n", "\n", "\"\"\"---设置文件保存路径---\"\"\"\n", "#目录和文件的分隔务必用/\n", - "directory = \"临时文件/批量生成题目/\"\n", - "filename = \"立体几何1批量.tex\"\n", + "directory = \"临时文件/\"\n", + "filename = \"赋能批量\"\n", "\"\"\"---设置文件名结束---\"\"\"\n", "if directory[-1] != \"/\":\n", " directory += \"/\"\n", diff --git a/工具/文本文件/metadata.txt b/工具/文本文件/metadata.txt index b7f89317..f1215e7b 100644 --- a/工具/文本文件/metadata.txt +++ b/工具/文本文件/metadata.txt @@ -1,41 +1,446 @@ -usages +tags +000275 +抛物线 -010540 -20221105 2023届高三02班 0.839 0.774 0.710 +000279 +抛物线 -010535 -20221105 2023届高三02班 1.000 1.000 1.000 +000337 +抛物线 -000222 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+抛物线 + +008933 +抛物线 + +008934 +抛物线 + +008935 +抛物线 + +008936 +抛物线 + +008937 +抛物线 + +008953 +抛物线 + +008954 +抛物线 + +008955 +抛物线 + +008956 +抛物线 + +008957 +抛物线 + +008960 +抛物线 + +008966 +抛物线 + +008968 +抛物线 + +009077 +抛物线 + +009081 +抛物线 + +009098 +抛物线 + +009099 +抛物线 + +009105 +抛物线 + +009106 +抛物线 + +009110 +抛物线 + +009836 +抛物线 + +009837 +抛物线 + +009838 +抛物线 + +009839 +抛物线 + +009840 +抛物线 + +010682 +抛物线 + +010683 +抛物线 + +010684 +抛物线 + +010685 +抛物线 + +010686 +抛物线 + +010688 +抛物线 + +010689 +抛物线 + +010703 +抛物线 -009744 -20221105 2023届高三02班 0.710 diff --git a/工具/文本文件/题号筛选.txt b/工具/文本文件/题号筛选.txt index 72b822bf..764c8a0a 100644 --- a/工具/文本文件/题号筛选.txt +++ b/工具/文本文件/题号筛选.txt @@ -1 +1 @@ -000291,000292,000293,000294,000296,000297,000299,000301,000302,000304,000305,000781,001944,001947,001948,001949,001950,001951,001952,001953,001954,001955,001956,001957,001958,001959,001960,001961,001962,001963,001964,001965,001966,001968,001969,001971,001972,001973,001974,001975,001976,001977,001978,001979,001980,001981,001985,001987,001991,003624,003647,003679,004096,004243,004348,004656,004698,004740,009855,009856,009857,009858,009859,009860,009861,009862,009863,009864,009865,009867,009868,009870,009871,009872,009873,010706,010707,010708,010709,010710,010711,010712,010713,010714,010715,010716,010717,010718,010719,010720,010721,010722,010723,010724,010725,010726,010727,010729,010730,010731,010732,010733,010735,010736,010737,010738,010739,010740,030452,030453,030454,030455,030456,030457,030458,030459,030460,030461,030462,030463,030464,030465,030466,030467,030468,030469,030470,030471,030472 \ No newline at end of file +000263,000267,000273,000282,000283,000286,000287,000290,000395,000484,000523,000544,000597,000610,000634,000825,000839,000865,000916,002095,002096,002097,002098,002099,002100,002101,002102,002103,002104,002105,002106,002108,002109,002110,002111,002113,002114,002115,002116,002117,002119,002120,002121,002122,002123,002124,002135,002153,002181,002192,002202,002203,002211,002220,002237,002238,002246,002292,002293,002294,002296,002297,002298,002299,002301,002312,002320,002340,002341,002355,002357,002359,002383,002388,002396,002399,002414,002444,002446,002449,002453,002455,002457,002458,002459,002460,002461,002462,002463,002464,002465,002466,002473,002474,002476,002477,002691,003371,003379,003387,003405,003425,003433,003442,003445,003602,003659,003663,003671,003740,003744,003756,003766,003780,003796,003804,003831,003861,003864,003917,003929,003933,003937,003944,003972,003973,003978,003980,003981,004158,004207,004237,004246,004260,004267,004287,004343,004504,004508,004650,004718,004734,004742,004749,008764,008782,008789,008836,008837,008838,008839,008840,008841,008842,008843,008844,008845,008846,008847,008848,008849,008850,008851,008852,008853,008857,008867,008874,008888,008896,008911,008918,008938,008967,009079,009082,009103,009108,009810,009811,009818,009826,009835,009841,009842,009843,009844,009847,009848,009851,009852,009853,009854,009999,010611,010625,010632,010633,010634,010641,010663,010665,010681,010687,010691,010692,010693,010694,010695,010696,010699,010705 \ No newline at end of file diff --git a/工具/添加题目到数据库.ipynb b/工具/添加题目到数据库.ipynb index f53f224e..0e7d5da8 100644 --- a/工具/添加题目到数据库.ipynb +++ b/工具/添加题目到数据库.ipynb @@ -2,20 +2,20 @@ "cells": [ { "cell_type": "code", - "execution_count": 5, + "execution_count": 2, "metadata": {}, "outputs": [], "source": [ "#修改起始id,出处,文件名\n", - "starting_id = 30452\n", - "origin = \"高中数学教与学例题与习题\"\n", + "starting_id = 12030\n", + "origin = \"2022届高三第一轮复习讲义\"\n", "filename = r\"C:\\Users\\weiye\\Documents\\wwy sync\\临时工作区\\自拟题目4.tex\"\n", - "editor = \"20221104\\t王伟叶\"" + "editor = \"20221105\\t王伟叶\"" ] }, { "cell_type": "code", - "execution_count": 6, + "execution_count": 3, "metadata": {}, "outputs": [], "source": [ diff --git a/工具/生成文件夹下的题号清单.ipynb b/工具/生成文件夹下的题号清单.ipynb new file mode 100644 index 00000000..84c3c85e --- /dev/null +++ b/工具/生成文件夹下的题号清单.ipynb @@ -0,0 +1,327 @@ +{ + "cells": [ + { + "cell_type": "code", + "execution_count": 63, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "周末卷01.tex\n", + "填空题\n", + "(010923)\n", + "(010924)\n", + "(010925)\n", + "(010926)\n", + "(030013)\n", + "(010928)\n", + "(010929)\n", + "(010930)\n", + "(030014)\n", + "(010932)\n", + "(010933)\n", + "(010934)\n", + "选择题\n", + "(010935)\n", + "(010936)\n", + "(010937)\n", + "(010938)\n", + "解答题\n", + "(010939)\n", + "(010940)\n", + "(010941)\n", + "(010942)\n", + "(010943)\n", + "\n", + "\n", + "\n", + "周末卷02.tex\n", + "填空题\n", + "(010944)\n", + "(030017)\n", + "(010946)\n", + "(010947)\n", + "(010948)\n", + "(010949)\n", + "(010950)\n", + "(010951)\n", + "(010952)\n", + "(010953)\n", + "(010954)\n", + "(010955)\n", + "选择题\n", + "(010956)\n", + "(002874)\n", + "(010958)\n", + "(010959)\n", + "解答题\n", + "(010960)\n", + "(010961)\n", + "(010962)\n", + "(010963)\n", + "(010964)\n", + "\n", + "\n", + "\n", + "周末卷03.tex\n", + "填空题\n", + "(003115)\n", + "(006264)\n", + "(003096)\n", + "(001472)\n", + "(006604)\n", + "(030027)\n", + "(030028)\n", + "选择题\n", + "(006305)\n", + "解答题\n", + "(006460)\n", + "(006463)\n", + "\n", + "\n", + "\n", + "周末卷03_暂未使用.tex\n", + "填空题\n", + "(010965)\n", + "(010966)\n", + "(030023)\n", + "(010968)\n", + "(010969)\n", + "(010970)\n", + "(030025)\n", + "(010972)\n", + "(030024)\n", + "(010974)\n", + "(010975)\n", + "(010976)\n", + "选择题\n", + "(010977)\n", + "(002745)\n", + "(010979)\n", + "(010980)\n", + "解答题\n", + "(010981)\n", + "(010982)\n", + "(010983)\n", + "(010984)\n", + "(010985)\n", + "\n", + "\n", + "\n", + "周末卷04.tex\n", + "填空题\n", + "(001853)\n", + "(030108)\n", + "(003355)\n", + "(000655)\n", + "(000724)\n", + "(001860)\n", + "(002038)\n", + "(030106)\n", + "(030107)\n", + "(003621)\n", + "选择题\n", + "(001846)\n", + "(002013)\n", + "(003703)\n", + "解答题\n", + "(001557)\n", + "(004702)\n", + "\n", + "\n", + "\n", + "周末卷05.tex\n", + "填空题\n", + "(030169)\n", + "(030273)\n", + "(001677)\n", + "(003531)\n", + "(003533)\n", + "(003455)\n", + "选择题\n", + "(004092)\n", + "(003891)\n", + "(001643)\n", + "解答题\n", + "(000182)\n", + "(000187)\n", + "(000298)\n", + "(003495)\n", + "(004180)\n", + "(003500)\n", + "(003462)\n", + "\n", + "\n", + "\n", + "周末卷06.tex\n", + "填空题\n", + "(010497)\n", + "(010501)\n", + "(001726)\n", + "(030279)\n", + "(030280)\n", + "(030278)\n", + "(001631)\n", + "(001668)\n", + "(001724)\n", + "选择题\n", + "(001676)\n", + "(010487)\n", + "(009998)\n", + "(010491)\n", + "解答题\n", + "(010470)\n", + "(010533)\n", + "(010508)\n", + "(010000)\n", + "\n", + "\n", + "\n", + "周末卷07.tex\n", + "填空题\n", + "(004446)\n", + "(004447)\n", + "(004448)\n", + "(004449)\n", + "(004450)\n", + "(004451)\n", + "(004453)\n", + "(004454)\n", + "(004455)\n", + "(004456)\n", + "(004457)\n", + "选择题\n", + "(004458)\n", + "解答题\n", + "(004462)\n", + "(004463)\n", + "(004464)\n", + "\n", + "\n", + "\n", + "国庆卷.tex\n", + "课前练习\n", + "(030033)\n", + "(030030)\n", + "(030032)\n", + "(030031)\n", + "(030029)\n", + "(030034)\n", + "(030076)\n", + "(030036)\n", + "(030037)\n", + "(030039)\n", + "(030044)\n", + "(030038)\n", + "(030040)\n", + "(030041)\n", + "(030045)\n", + "(030042)\n", + "(030048)\n", + "(030046)\n", + "(030047)\n", + "(030051)\n", + "(030043)\n", + "(030052)\n", + "(030050)\n", + "(030049)\n", + "(030053)\n", + "(030054)\n", + "(030055)\n", + "(030059)\n", + "(030058)\n", + "(030056)\n", + "(030057)\n", + "(030060)\n", + "(030061)\n", + "(030063)\n", + "(030062)\n", + "(030065)\n", + "(030067)\n", + "(030066)\n", + "(030064)\n", + "(030068)\n", + "\n", + "\n", + "\n" + ] + } + ], + "source": [ + "import os,re\n", + "\"---此处输入文件夹名---\"\n", + "directory = r\"C:\\Users\\weiye\\Documents\\wwy sync\\23届\\上学期周末卷\"\n", + "\"---文件夹名输入结束---\"\n", + "\n", + "filelist = [filename for filename in os.listdir(directory) if \".tex\" in filename]\n", + "\n", + "output = \"\"\n", + "\n", + "for filename in filelist:\n", + " print(filename)\n", + " output += filename + \"\\n\"\n", + " with open(os.path.join(directory,filename),\"r\",encoding = \"u8\") as f:\n", + " try:\n", + " data = re.findall(r\"\\\\begin{document}([\\s\\S]*?)\\\\end{document}\",f.read())[0]\n", + " data = data.replace(r\"\\section\",\"endsecbeginsec\") + \"endsec\"\n", + " sectionlist = re.findall(r\"beginsec([\\s\\S]*?)endsec\",data)\n", + " for sec in sectionlist:\n", + " secname = re.findall(r\"{([\\S]*)}\",sec)[0]\n", + " output += secname + \"\\n\"\n", + " print(secname)\n", + " for id in re.findall(r\"\\(\\d{6}\\)\",sec):\n", + " print(id)\n", + " output += id + \"\\n\"\n", + " except:\n", + " pass \n", + " output += \"\\n\\n\"\n", + " print(\"\\n\\n\")\n", + "\n", + "with open(os.path.join(directory,\"题号清点.txt\"),\"w\",encoding = \"u8\") as f:\n", + " f.write(output + \"\\n\\n\\n以下题号不含括号\\n\\n\\n\" + output.replace(\"(\",\"\").replace(\")\",\"\"))\n", + "\n", + " \n", + "\n", + "\n", + " \n", + " \n" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [] + } + ], + "metadata": { + "kernelspec": { + "display_name": "Python 3.8.8 ('base')", + "language": "python", + "name": "python3" + }, + "language_info": { + "codemirror_mode": { + "name": "ipython", + "version": 3 + }, + "file_extension": ".py", + "mimetype": "text/x-python", + "name": "python", + "nbconvert_exporter": "python", + "pygments_lexer": "ipython3", + "version": "3.8.8" + }, + "orig_nbformat": 4, + "vscode": { + "interpreter": { + "hash": "d311ffef239beb3b8f3764271728f3972d7b090c974f8e972fcdeedf230299ac" + } + } + }, + "nbformat": 4, + "nbformat_minor": 2 +} diff --git a/工具/识别题库中尚未标注的题目类型.ipynb b/工具/识别题库中尚未标注的题目类型.ipynb index 4abc7ea5..8e7f957a 100644 --- a/工具/识别题库中尚未标注的题目类型.ipynb +++ b/工具/识别题库中尚未标注的题目类型.ipynb @@ -9,29 +9,30 @@ "name": "stdout", "output_type": "stream", "text": [ - "030428 填空题\n", - "030429 填空题\n", - "030430 填空题\n", - "030431 填空题\n", - "030432 填空题\n", - "030433 填空题\n", - "030434 填空题\n", - "030435 填空题\n", - "030436 填空题\n", - "030437 填空题\n", - "030438 填空题\n", - "030439 填空题\n", - "030440 选择题\n", - "030441 选择题\n", - "030442 选择题\n", - "030443 选择题\n", - "030444 解答题\n", - "030445 解答题\n", - "030446 解答题\n", - "030447 解答题\n", - "030448 解答题\n", - "030449 填空题\n", - "030450 填空题\n" + "012030 填空题\n", + "012031 填空题\n", + "012032 填空题\n", + "030452 解答题\n", + "030453 解答题\n", + "030454 填空题\n", + "030455 选择题\n", + "030456 解答题\n", + "030457 解答题\n", + "030458 解答题\n", + "030459 解答题\n", + "030460 解答题\n", + "030461 解答题\n", + "030462 解答题\n", + "030463 填空题\n", + "030464 填空题\n", + "030465 填空题\n", + "030466 填空题\n", + "030467 选择题\n", + "030468 选择题\n", + "030469 解答题\n", + "030470 填空题\n", + "030471 选择题\n", + "030472 解答题\n" ] } ], diff --git a/工具/题号选题pdf生成.ipynb b/工具/题号选题pdf生成.ipynb index ecb39c14..766ba0f2 100644 --- a/工具/题号选题pdf生成.ipynb +++ b/工具/题号选题pdf生成.ipynb @@ -2,16 +2,16 @@ "cells": [ { "cell_type": "code", - "execution_count": 4, + "execution_count": 10, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ - "开始编译教师版本pdf文件: 临时文件/立体几何空间向量_教师用_20221104.tex\n", + "开始编译教师版本pdf文件: 临时文件/圆预选_教师用_20221105.tex\n", "0\n", - "开始编译学生版本pdf文件: 临时文件/立体几何空间向量_学生用_20221104.tex\n", + "开始编译学生版本pdf文件: 临时文件/圆预选_学生用_20221105.tex\n", "0\n" ] } @@ -35,7 +35,7 @@ "\n", "\"\"\"---设置文件名---\"\"\"\n", "#目录和文件的分隔务必用/\n", - "filename = \"临时文件/立体几何空间向量\"\n", + "filename = \"临时文件/圆预选\"\n", "\"\"\"---设置文件名结束---\"\"\"\n", "\n", "\n", diff --git a/题库0.3/Problems.json b/题库0.3/Problems.json index 9937f5d9..36ab5394 100644 --- a/题库0.3/Problems.json +++ b/题库0.3/Problems.json @@ -6335,7 +6335,8 @@ "K0701002X" ], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -6358,7 +6359,8 @@ "K0707001X" ], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -6382,7 +6384,8 @@ "K0707001X" ], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -6405,7 +6408,8 @@ "K0707005X" ], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -6430,7 +6434,8 @@ "K0701002X" ], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -6454,7 +6459,8 @@ "K0702002X" ], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -6477,7 +6483,8 @@ "K0708003X" ], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -6502,7 +6509,8 @@ "K0705003X" ], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -6526,7 +6534,8 @@ "K0707001X" ], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -6550,7 +6559,8 @@ "K0702002X" ], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -6574,7 +6584,8 @@ "K0705003X" ], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -6598,7 +6609,8 @@ "K0707001X" ], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -6624,7 +6636,8 @@ "K0706002X" ], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -6648,7 +6661,8 @@ "K0706003X" ], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -6671,7 +6685,8 @@ "K0707005X" ], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -6694,7 +6709,8 @@ "K0703002X" ], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -6718,7 +6734,8 @@ "K0706001X" ], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -6743,7 +6760,8 @@ "K0708003X" ], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -6767,7 +6785,8 @@ "K0708003X" ], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -6791,7 +6810,8 @@ "K0702001X" ], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -6838,7 +6858,8 @@ "K0701002X" ], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -6861,7 +6882,8 @@ "K0706002X" ], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -6884,7 +6906,8 @@ "K0708003X" ], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -6978,7 +7001,9 @@ "K0713004X" ], "tags": [ - "第七单元" + "第七单元", + "椭圆", + "双曲线" ], "genre": "解答题", "ans": "", @@ -7003,7 +7028,8 @@ "K0714005X" ], "tags": [ - "第七单元" + "第七单元", + "椭圆" ], "genre": "解答题", "ans": "", @@ -7027,7 +7053,8 @@ "K0711002X" ], "tags": [ - "第七单元" + "第七单元", + "双曲线" ], "genre": "解答题", "ans": "", @@ -7099,7 +7126,8 @@ "K0719006X" ], "tags": [ - "第七单元" + "第七单元", + "抛物线" ], "genre": "解答题", "ans": "", @@ -7199,7 +7227,8 @@ "K0721002X" ], "tags": [ - "第七单元" + "第七单元", + "抛物线" ], "genre": "解答题", "ans": "", @@ -7222,7 +7251,8 @@ "K0716002X" ], "tags": [ - "第七单元" + "第七单元", + "双曲线" ], "genre": "解答题", "ans": "", @@ -7324,7 +7354,8 @@ "K0715003X" ], "tags": [ - "第七单元" + "第七单元", + "椭圆" ], "genre": "解答题", "ans": "", @@ -7347,7 +7378,8 @@ "K0715002X" ], "tags": [ - "第七单元" + "第七单元", + "椭圆" ], "genre": "解答题", "ans": "", @@ -7417,7 +7449,9 @@ "K0716003X" ], "tags": [ - "第七单元" + "第七单元", + "椭圆", + "双曲线" ], "genre": "解答题", "ans": "", @@ -7441,7 +7475,8 @@ "K0717002X" ], "tags": [ - "第七单元" + "第七单元", + "双曲线" ], "genre": "解答题", "ans": "", @@ -8673,7 +8708,8 @@ "content": "已知直线$l$经过点$(-\\sqrt{5},0)$且方向向量为$(2,-1)$, 则原点$O$到直线$l$的距离为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "填空题", "ans": "$1$", @@ -8698,7 +8734,8 @@ "content": "若双曲线的一条渐近线为$x+2y=0$, 且双曲线与抛物线$y=x^2$的准线仅有一个公共点, 则此双曲线的标准方程为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "双曲线" ], "genre": "填空题", "ans": "$16y^2-4x^2=1$", @@ -8772,7 +8809,8 @@ "content": "已知抛物线$C$的顶点在平面直角坐标系原点, 焦点在$x$轴上, 若$C$经过点$M(1,3)$, 则其焦点到准线的距离为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "抛物线" ], "genre": "填空题", "ans": "$\\dfrac 92$", @@ -9578,7 +9616,8 @@ "content": "设$k\\in \\mathbf{R}$, $\\dfrac{y^2}{k}-\\dfrac{x^2}{k-2}=1$表示焦点在$y$轴上的双曲线, 则半焦距的取值范围是\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "双曲线" ], "genre": "填空题", "ans": "$(\\sqrt 2,+\\infty)$", @@ -9908,7 +9947,8 @@ "content": "抛物线$y=x^2$上一点$M$到焦点的距离为$1$, 则点$M$的纵坐标为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "抛物线" ], "genre": "填空题", "ans": "$\\frac 34$", @@ -10265,7 +10305,8 @@ "content": "椭圆$\\begin{cases} x=5\\cos \\theta, \\\\ y=4\\sin \\theta \\end{cases}$($\\theta$为参数)的焦距为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "椭圆" ], "genre": "填空题", "ans": "$6$", @@ -10335,7 +10376,7 @@ }, "000382": { "id": "000382", - "content": "已知向量$\\overrightarrow{a}=(1,2)$, $\\overrightarrow{b}=(0,3)$, 则$\\overrightarrow{b}$在$\\overrightarrow{a}$的方向上的投影为\\blank{50}.", + "content": "已知向量$\\overrightarrow{a}=(1,2)$, $\\overrightarrow{b}=(0,3)$, 则$\\overrightarrow{b}$在$\\overrightarrow{a}$的方向上的数量投影为\\blank{50}.", "objs": [], "tags": [ "第五单元" @@ -10356,7 +10397,8 @@ ], "origin": "赋能练习", "edit": [ - "20220624\t朱敏慧, 王伟叶" + "20220624\t朱敏慧, 王伟叶", + "20221105\t王伟叶" ], "same": [], "related": [ @@ -10367,7 +10409,7 @@ }, "000383": { "id": "000383", - "content": "已知一个底面置于水平面上的圆锥, 其左视图是边长为6的正三角形, 则该圆锥的侧面积为\\blank{50}.", + "content": "已知一个底面置于水平面上的圆锥, 其轴截面是边长为$6$的正三角形, 则该圆锥的侧面积为\\blank{50}.", "objs": [], "tags": [ "暂无对应" @@ -10381,7 +10423,8 @@ ], "origin": "赋能练习", "edit": [ - "20220624\t朱敏慧, 王伟叶" + "20220624\t朱敏慧, 王伟叶", + "20221105\t王伟叶" ], "same": [], "related": [], @@ -10796,7 +10839,8 @@ "content": "等轴双曲线$x^2-y^2=a^2$与抛物线$y^2=16x$的准线交于$A$、$B$两点, 且$|AB|=4\\sqrt3$, 则该双曲线的实轴长等于\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "双曲线" ], "genre": "填空题", "ans": "$4$", @@ -10901,7 +10945,7 @@ }, "000403": { "id": "000403", - "content": "已知数列$\\{a_n\\}$的通项公式为$a_n=n^2+bn$, 若数列$\\{a_n\\}$是单调递增数列, 则实数$b$的取值范围是\\blank{50}.", + "content": "已知数列$\\{a_n\\}$的通项公式为$a_n=n^2+bn$, 若数列$\\{a_n\\}$是严格递增数列, 则实数$b$的取值范围是\\blank{50}.", "objs": [ "K0406004X" ], @@ -10918,7 +10962,8 @@ ], "origin": "赋能练习", "edit": [ - "20220624\t朱敏慧, 王伟叶" + "20220624\t朱敏慧, 王伟叶", + "20221105\t王伟叶" ], "same": [], "related": [], @@ -11454,7 +11499,8 @@ "content": "过双曲线$C:\\dfrac{x^2}{a^2}-\\dfrac{y^2}4=1$的右焦点$F$作一条垂直于$x$轴的垂线交双曲线$C$的两条渐近线于$A$、$B$两点, $O$为坐标原点, 则$\\triangle OAB$的面积的最小值为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "双曲线" ], "genre": "填空题", "ans": "$8$", @@ -11675,7 +11721,8 @@ "content": "若双曲线$x^2-\\dfrac{y^2}{b^2}=1$的一个焦点到其渐近线距离为$2\\sqrt2$, 则该双曲线焦距等于\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "双曲线" ], "genre": "填空题", "ans": "$6$", @@ -12157,7 +12204,8 @@ "content": "点$(1,0)$到双曲线$\\dfrac{x^2}4-y^2=1$的渐近线的距离是\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "双曲线" ], "genre": "填空题", "ans": "$\\frac{\\sqrt 5}5$", @@ -12501,7 +12549,8 @@ "content": "已知抛物线$C$的顶点为坐标原点, 双曲线$\\dfrac{x^2}{25}-\\dfrac{y^2}{144}=1$的右焦点是$C$的焦点$F$. 若斜率为$-1$, 且过$F$的直线与$C$交于$A,B$两点, 则$|AB|=$\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "抛物线" ], "genre": "填空题", "ans": "$104$", @@ -12573,7 +12622,8 @@ "content": "抛物线$y^2=4x$的焦点坐标是\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "抛物线" ], "genre": "填空题", "ans": "$(1,0)$", @@ -13070,7 +13120,8 @@ "content": "设焦点为$F_1$、$F_2$的椭圆$\\dfrac{x^2}{a^2}+\\dfrac{y^2}3=1 \\ (a>0)$上的一点$P$也在抛物线$y^2=\\dfrac94x$上, 抛物线焦点为$F_3$, 若$|PF_3|=\\dfrac{25}{16}$, 则$\\triangle PF_1F_2$的面积为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "椭圆" ], "genre": "填空题", "ans": "$\\dfrac 32$", @@ -13274,7 +13325,8 @@ "content": "在平面直角坐标系中, 双曲线$\\dfrac{x^2}{a^2}-y^2=1 $的一个顶点与抛物线$y^2=12x$的焦点重合, 则双曲线的两条渐近线的方程为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "双曲线" ], "genre": "填空题", "ans": "$y=\\pm \\frac 13x$", @@ -13718,10 +13770,11 @@ }, "000510": { "id": "000510", - "content": "已知$F_1$、$F_2$是椭圆$\\dfrac{x^2}{25}+\\dfrac{y^2}9=1$的两个焦点, $P$是椭圆上的一个动点, 则$|PF1|\\times |PF2|$的最大值是\\blank{50}.", + "content": "已知$F_1$、$F_2$是椭圆$\\dfrac{x^2}{25}+\\dfrac{y^2}9=1$的两个焦点, $P$是椭圆上的一个动点, 则$|PF_1|\\times |PF_2|$的最大值是\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "椭圆" ], "genre": "填空题", "ans": "$25$", @@ -13804,7 +13857,8 @@ "content": "已知点$A(2,3)$、点$B(-2,\\sqrt3)$, 直线$l$过点$P(-1,0)$, 若直线$l$与线段$AB$相交, 则直线$l$的倾斜角的取值范围是\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "填空题", "ans": "$[\\frac{\\pi }4, \\frac{2\\pi }3]$", @@ -13958,7 +14012,8 @@ "content": "与双曲线$\\dfrac{x^2}9-\\dfrac{y^2}{16}=1$的渐近线相同, 且经过点$A(-3,2 \\sqrt3)$的双曲线的方程是\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "双曲线" ], "genre": "填空题", "ans": "$\\frac{4{x^2}}9-\\frac{y^2}4=1$", @@ -14059,7 +14114,8 @@ "content": "已知点$A(2,3)$到直线$ax+(a-1)y+3=0$的距离不小于$3$, 则实数$a$的取值范围是\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "填空题", "ans": "$(-\\infty ,-3 ]\\cup [ \\frac 37,+\\infty)$", @@ -14260,7 +14316,8 @@ "content": "已知直线$l$的一个法向量是$\\overrightarrow{n}=(\\sqrt3,-1)$, 则$l$的倾斜角的大小是\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "填空题", "ans": "$\\frac{\\pi }3$", @@ -15038,7 +15095,9 @@ "content": "在平面直角坐标系$xOy$中, 以直线$y=\\pm 2x$为渐近线, 且经过椭圆$x^2+\\dfrac{y^2}4=1$右顶点的双曲线的方程是\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "椭圆", + "双曲线" ], "genre": "填空题", "ans": "$x^2-\\dfrac{y^2}4=1$", @@ -15667,7 +15726,8 @@ "content": "抛物线$y^2=-8x$的焦点与双曲线$\\dfrac{x^2}{a^2}-y^2=1$的左焦点重合, 则这条双曲线的两条渐近线的夹角为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "双曲线" ], "genre": "填空题", "ans": "$\\dfrac{\\pi }3$", @@ -16574,7 +16634,8 @@ "content": "直线$\\begin{cases} x=-2-\\sqrt2 t, \\\\y=3+\\sqrt2 t, \\end{cases}$($t$为参数)对应的普通方程是\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "填空题", "ans": "$x+y-1=0$", @@ -16733,7 +16794,8 @@ "content": "已知椭圆$x^2+\\dfrac{y^2}{b^2}=1\\ (00)$,它的渐近线方程是$y=\\pm 2x$,则$a$的值为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "双曲线" ], "genre": "填空题", "ans": "$2$", @@ -17214,7 +17278,8 @@ "content": "在平面直角坐标系中, 已知点$P(-2,2)$, 对于任意不全为零的实数$a$、$b$, 直线$l:a(x-1)+b(y+2)=0$, 若点$P$到直线$l$的距离为$d$, 则$d$的取值范围是\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "填空题", "ans": "$[0,5]$", @@ -17684,7 +17749,8 @@ "content": "设$P,Q$分别为直线$\\begin{cases} x=t, \\\\ y=6-2t, \\end{cases}$($t$为参数)和曲线$C:\\begin{cases} x=1+\\sqrt5\\cos\\theta, \\\\ y=-2+\\sqrt5\\sin\\theta,\\end{cases}$($\\theta$为参数)的点, 则$|PQ|$的最小值为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "填空题", "ans": "$\\dfrac{\\sqrt 5}5$", @@ -17835,7 +17901,8 @@ "content": "若直线$l$的参数方程为$\\begin{cases} x=4-4t, \\\\ y=-2+3t,\\end{cases} \\ t\\in \\mathbf{R}$, 则直线$l$在$y$轴上的截距是\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "填空题", "ans": "$1$", @@ -17885,7 +17952,8 @@ "content": "抛物线$y=\\dfrac14{x^2}$的焦点到准线的距离为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "抛物线" ], "genre": "填空题", "ans": "$2$", @@ -18238,7 +18306,8 @@ "content": "直线$\\begin{cases} x=2+t, \\\\ y=4-t,\\end{cases}$($t$为参数)与曲线$\\begin{cases} x=3+\\sqrt2\\cos\\theta, \\\\ y=5+\\sqrt2\\sin\\theta \\end{cases}$($\\theta$为参数)的公共点的个数是\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "填空题", "ans": "$1$", @@ -18265,7 +18334,8 @@ "content": "已知双曲线$C_1$与双曲线$C_2$的焦点重合,$C_1$的方程为$\\dfrac{x^2}3-{y^2}=1$, 若$C_2$的一条渐近线的倾斜角是$C_1$的一条渐近线的倾斜角的$2$倍, 则$C_2$的方程为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "双曲线" ], "genre": "填空题", "ans": "$x^2-\\dfrac{y^2}3=1$", @@ -18434,7 +18504,8 @@ "content": "已知双曲线$\\dfrac{x^2}{a^2}-\\dfrac{y^2}{(a+3)^2}=1 \\ (a>0)$的一条渐近线方程为$y=\\pm 2x$, 则$a=$\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "双曲线" ], "genre": "填空题", "ans": "$3$", @@ -18516,7 +18587,8 @@ "content": "直线$\\begin{cases} x=t-1, \\\\ y=2-t,\\end{cases}$($t$为参数)与曲线$\\begin{cases} x=3\\cos\\theta, \\\\ y=2\\sin\\theta,\\end{cases}$($\\theta$为参数)的交点个数是\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "填空题", "ans": "$2$", @@ -18860,7 +18932,8 @@ "content": "设$A$是椭圆$\\dfrac{x^2}{a^2}+\\dfrac{y^2}{a^2-4}=1 \\ (a>0)$上的动点, 点$F$的坐标为$(-2,0)$, 若满足$|AF|=10$的点$A$有且仅有两个, 则实数$a$的取值范围为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "椭圆" ], "genre": "填空题", "ans": "$(8,12)$", @@ -18912,7 +18985,8 @@ "content": "设抛物线的焦点坐标为$(1,0)$, 则此抛物线的标准方程为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "抛物线" ], "genre": "填空题", "ans": "$y^2=4x$", @@ -19338,7 +19412,8 @@ "content": "已知椭圆$\\dfrac{x^2}{a^2}+y^2=1 \\ (a>0)$的焦点$F_1$、$F_2$, 抛物线${y^2}=2x$的焦点为$F$, 若$\\overrightarrow{F_1F}=3 \\overrightarrow{FF_2}$, 则$a=$\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "椭圆" ], "genre": "填空题", "ans": "$\\sqrt 2$", @@ -19470,7 +19545,8 @@ "content": "抛物线$y=x^2$的焦点坐标是\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "抛物线" ], "genre": "填空题", "ans": "$(0,\\frac 14)$", @@ -19765,7 +19841,8 @@ "content": "已知抛物线$x^2=ay$的准线方程是$y=-\\dfrac14$, 则$a=$\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "抛物线" ], "genre": "填空题", "ans": "$1$", @@ -19911,7 +19988,8 @@ "content": "已知直线$l_1:mx-y=0$, $l_2:x+my-m-2=0$. 当$m$在实数范围内变化时, $l_1$与$l_2$的交点$P$恒在一个定圆上, 则定圆方程是\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "填空题", "ans": "$x^2+y^2-2x-y=0$", @@ -19960,7 +20038,8 @@ "content": "直线$ax+(a-1)y+1=0$与直线$4x+ay-2=0$互相平行, 则实数$a=$\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "填空题", "ans": "$2$", @@ -20063,7 +20142,8 @@ "content": "从集合$\\{-1,1,2,3\\}$随机取一个为$m$, 从集合$\\{-2,-1,1,2\\}$随机取一个为$n$, 则方程$\\dfrac{x^2}m+\\dfrac{y^2}n=1$表示双曲线的概率为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "双曲线" ], "genre": "填空题", "ans": "$\\frac 12$", @@ -20162,7 +20242,8 @@ "content": "椭圆的长轴长等于$m$, 短轴长等于$n$, 则此椭圆的内接矩形的面积的最大值为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "椭圆" ], "genre": "填空题", "ans": "$\\frac 12mn$", @@ -20706,7 +20787,8 @@ "content": "平面上三条直线$x-2y+1=0$, $x-1=0$, $x+ky=0$, 如果这三条直线将平面划分为六个部分, 则实数$k$的取值组成的集合$A=$\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "填空题", "ans": "$\\{–1, 0, –2\\}$", @@ -20921,7 +21003,8 @@ "content": "已知抛物线的顶点在坐标原点, 焦点在$y$轴上, 抛物线上一点$M(a,-4) \\ (a>0)$到焦点F的距离为$5$. 则该抛物线的标准方程为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "抛物线" ], "genre": "填空题", "ans": "$x^2=-4y$", @@ -20995,7 +21078,8 @@ "content": "双曲线$\\dfrac{x^2}{a^2}-\\dfrac{y^2}9=1 \\ (a>0)$的渐近线方程为$3x\\pm 2y=0$, 则$a=$\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "双曲线" ], "genre": "填空题", "ans": "$2$", @@ -21110,7 +21194,8 @@ "content": "直线$l$的参数方程为$\\begin{cases} x=1+t, \\\\ y=-1+2t, \\end{cases}$($t$为参数), 则$l$的一个法向量为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "填空题", "ans": "$(2,-1)$(不唯一)", @@ -21425,7 +21510,8 @@ "content": "椭圆$\\begin{cases} x=2 \\cos\\theta, \\\\ y=\\sqrt3\\sin\\theta \\end{cases}$($\\theta$为参数)的右焦点为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "椭圆" ], "genre": "填空题", "ans": "$(1,0)$", @@ -21496,7 +21582,8 @@ "content": "已知抛物线型拱桥的顶点距水面$2$米时, 量得水面宽为$8$米. 当水面下降$1$米后, 水面的宽为\\blank{50}米.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "抛物线" ], "genre": "填空题", "ans": "$4\\sqrt 6$", @@ -21544,7 +21631,8 @@ "content": "抛物线$x^2=12y$的准线方程为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "抛物线" ], "genre": "填空题", "ans": "$y=-3$", @@ -21722,7 +21810,8 @@ "content": "在平面直角坐标系$xOy$中, 直线$l$的参数方程为$\\begin{cases} x=\\dfrac{\\sqrt2}2t-\\sqrt2, \\\\ y=\\dfrac{\\sqrt2}4t, \\end{cases}$($t$为参数), 椭圆$C$的参数方程为$\\begin{cases} x=\\cos \\theta, \\\\ y=\\dfrac12\\sin \\theta, \\end{cases}$($\\theta$为参数), 则直线$l$与椭圆$C$的公共点坐标为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "椭圆" ], "genre": "填空题", "ans": "$(-\\frac{\\sqrt 2}2,\\frac{\\sqrt 2}4)$", @@ -22264,7 +22353,9 @@ "content": "若双曲线$\\dfrac{x^2}3-\\dfrac{16y^2}{p^2}=1\\ (p>0)$的左焦点在抛物线$y^2=2px$的准线上, 则$p=$\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "双曲线", + "抛物线" ], "genre": "填空题", "ans": "$4$", @@ -22657,7 +22748,8 @@ "content": "双曲线$2 x^2-y^2=6$的焦距为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "双曲线" ], "genre": "填空题", "ans": "$6$", @@ -23068,7 +23160,8 @@ "content": "在平面直角坐标系$xOy$中, 有一定点$A(1,1)$, 若线段$OA$的垂直平分线过抛物线$C:y^2=2px \\ (p>0)$的焦点, 则抛物线$C$的方程为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "抛物线" ], "genre": "填空题", "ans": "$y^2=4x$", @@ -23388,7 +23481,8 @@ "content": "已知双曲线$x^2-\\dfrac{y^2}4=1$的右焦点为$F$, 过点$F$且平行于双曲线的一条渐近线的直线与双曲线交于点$P$, $M$在直线$PF$上, 且满足$\\overrightarrow{OM}\\cdot \\overrightarrow{PF}=0$, 则$\\dfrac{|\\overrightarrow{PM}|}{|\\overrightarrow{PF}|}=$\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "双曲线" ], "genre": "填空题", "ans": "$\\frac 12$", @@ -23434,7 +23528,8 @@ "content": "抛物线$y^2=4x$的焦点坐标是\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "抛物线" ], "genre": "填空题", "ans": "$(1,0)$", @@ -23496,7 +23591,8 @@ "content": "若$\\overrightarrow d=(3,2)$是直线$l$的一个方向向量, 则$l$的倾斜角的大小为\\blank{50}(结果用反三角函数值表示).", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "填空题", "ans": "$\\arctan \\frac 23$", @@ -23898,7 +23994,8 @@ "content": "已知$F_1,F_2$是椭圆$C:\\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1\\ (a>b>0)$的两个焦点, $P$为椭圆上一点, 且$\\overrightarrow{PF_1}\\perp \\overrightarrow{PF_2}$, 若$\\triangle PF_1F_2$的面积为$9$, 则$b=$\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "椭圆" ], "genre": "填空题", "ans": "$3$", @@ -24231,7 +24328,8 @@ "content": "如图, $A$、$B$为椭圆$\\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1 \\ (a>b>0)$的两个顶点, 过椭圆的右焦点$F$作$x$轴的垂线, 与其交于点$C$. 若$AB\\parallel OC$($O$为坐标原点), 则直线$AB$的斜率为\\blank{50}.\n\\begin{center}\n \\begin{tikzpicture}[scale = 1.4]\n \\draw [->] (-2,0) -- (2,0) node [below] {$x$};\n \\draw [->] (0,-1.4) -- (0,1.4) node [left] {$y$};\n \\draw (0,0) node [below left] {$O$};\n \\draw (0,0) ellipse ({sqrt(2)} and 1);\n \\draw (1,0) node [below] {$F$} -- (1,{sqrt(2)/2}) node [above] {$C$} -- (0,0);\n \\draw ({-sqrt(2)},0) node [below left] {$A$} -- (0,1) node [above right] {$B$};\n \\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [ - "第七单元" + "第七单元", + "椭圆" ], "genre": "填空题", "ans": "$\\frac{\\sqrt 2}2$", @@ -24254,7 +24352,8 @@ "content": "若经过抛物线 $y^2=4x$焦点的直线$l$与圆$(x-4)^2+y^2=4$相切, 则直线$l$的方程为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "抛物线" ], "genre": "填空题", "ans": "$x\\pm \\frac{\\sqrt 5}2y-1=0$", @@ -24654,7 +24753,8 @@ "content": "已知双曲线$\\dfrac{x^2}{a^2}-\\dfrac{y^2}{b^2}=1 \\ (a>0,\\ b>0)$的一条渐近线方程是$y=\\sqrt3x$, 它的一个焦点与抛物线$y^2=16x$的焦点相同, 则双曲线的标准方程为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "双曲线" ], "genre": "填空题", "ans": "$\\frac{x^2}4-\\frac{y^2}{12}=1$", @@ -25205,7 +25305,8 @@ "content": "直线$x+2y-1=0$与直线$y=1$的夹角大小为\\blank{50}(结果用反三角函数值表示).", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "填空题", "ans": "$\\arccos \\frac{2\\sqrt 5}5$", @@ -25456,7 +25557,8 @@ "content": "双曲线$4 x^2-y^2=1$的一条渐近线与直线$tx+y+1=0$垂直, 则$t=$\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "双曲线" ], "genre": "填空题", "ans": "$\\pm \\frac 12$", @@ -25481,7 +25583,8 @@ "content": "已知抛物线$y^2=4x$上一点$M(x_0,2 \\sqrt3)$, 则点$M$到抛物线焦点的距离为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "抛物线" ], "genre": "填空题", "ans": "$4$", @@ -25554,7 +25657,8 @@ "content": "在平面直角坐标系$xOy$中, 将点$A(2,1)$绕原点$O$逆时针旋转$\\dfrac\\pi 4$到点$B$, 若直线$OB$的倾斜角为$\\alpha$, 则$\\cos \\alpha$的值为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "填空题", "ans": "$\\frac{\\sqrt{10}}{10}$", @@ -25748,7 +25852,8 @@ "content": "抛物线$y^2=x$上一点$M$到焦点的距离为$1$, 则点$M$的横坐标是\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "抛物线" ], "genre": "填空题", "ans": "$\\dfrac{3}{4}$", @@ -25846,7 +25951,8 @@ "content": "已知双曲线$x^2-\\dfrac{y^2}{m^2}=1 \\ (m>0)$的渐近线与圆$x^2+(y+2)^2=1$没有公共点, 则该双曲线的焦距的取值范围为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "双曲线" ], "genre": "填空题", "ans": "$(2,4)$", @@ -56502,7 +56608,8 @@ "content": "若点$M$到$x$轴的距离和它到直线$y=\\sqrt{3}x$的距离相等, 则点$M$的轨迹方程为\\blank{100}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "填空题", "ans": "", @@ -56622,7 +56729,8 @@ "content": "平面上动点$M$到点$P(1,0)$与直线$l: x=-1$的距离相等, 求动点$M$的轨迹方程.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -56937,7 +57045,8 @@ "content": "若直线经过点$(2,-3)$, 且平行于向量$\\overrightarrow{d}=(3,4)$. 则直线$l$的点方向式方程为\\blank{100}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "填空题", "ans": "", @@ -56964,7 +57073,8 @@ "content": "过点$(-1,0)$且与直线$\\dfrac{x+1}{5}=\\dfrac{y+1}{-3}$有相同的方向向量的直线方程为\\blank{100}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "填空题", "ans": "", @@ -56988,7 +57098,8 @@ "content": "将直线$2x-3y+4=0$写成点方向式方程, 你的结果是\\blank{100}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "填空题", "ans": "", @@ -57016,7 +57127,8 @@ "content": "直线$2x-3y-1=0$的一个方向向量为\\blank{100}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "填空题", "ans": "", @@ -57042,7 +57154,8 @@ "content": "直线$3x+2=0$的一个方向向量为\\blank{50},\n直线$4-3y=0$\\underline{所有的}方向向量为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "填空题", "ans": "", @@ -57068,7 +57181,8 @@ "content": "将直线$2x-3y+4=0$写成点方向式方程, 你的结果是\\blank{100}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "填空题", "ans": "", @@ -57096,7 +57210,8 @@ "content": "若直线$l$与两坐标轴围成一个等腰直角三角形, 则直线$l$的一个方向向量为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "填空题", "ans": "", @@ -57120,7 +57235,8 @@ "content": "已知直线$l$过点$(1,2)$, 且$M(2,3)$与$N(4,-5)$两点到直线$l$的距离相等. 则直线$l$的点方向式方程为\\blank{100}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "填空题", "ans": "", @@ -57144,7 +57260,8 @@ "content": "已知平行四边形$ABCD$的三个顶点的坐标分别为$A(1,2),B(3,4),C(2,6)$, 分别求$AB$边与$AD$边所在直线的点方向式方程.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -57170,7 +57287,8 @@ "content": "已知梯形$ABCD$的三个顶点的坐标分别为$A(2,3),B(-2,1),C(4,5)$, 求此梯形中位线所在直线的方程.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -57220,7 +57338,8 @@ "content": "若直线经过点$(2,-3)$, 且垂直于向量$(3,4)$. 则直线$l$的点法向式方程为\\blank{100}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "填空题", "ans": "", @@ -57247,7 +57366,8 @@ "content": "若直线经过点$(2,-3)$, 且垂直于向量$(3,4)$. 则直线$l$的点方向式方程为\\blank{100}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "填空题", "ans": "", @@ -57274,7 +57394,8 @@ "content": "将直线$2x-3y+4=0$写成点法向式方程, 你的结果是\\blank{100}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "填空题", "ans": "", @@ -57301,7 +57422,8 @@ "content": "直线$2x-3y-1=0$的一个法向量为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "填空题", "ans": "", @@ -57327,7 +57449,8 @@ "content": "直线$3x+2=0$的一个法向量为\\blank{50},\n直线$4-3y=0$\\underline{所有的}法向量为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "填空题", "ans": "", @@ -57353,7 +57476,8 @@ "content": "若直线$2(x+1)+9(y-1)=0$的一个法向量为$(a-1,a^2)$, 则实数$a$的值为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "填空题", "ans": "", @@ -57377,7 +57501,8 @@ "content": "已知原点$O$在直线$l$上的射影为$H(2,3)$, 则直线$l$的方程为\\blank{100}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "填空题", "ans": "", @@ -57401,7 +57526,8 @@ "content": "已知正方形$ABCD$的顶点$A(-1,1)$, 正方形中心坐标为$(0,3)$, 则对角线$BD$所在直线的方程为\\blank{100}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "填空题", "ans": "", @@ -57425,7 +57551,8 @@ "content": "过点$A(-1,1)$, 且与点$B(2,5)$距离最大的直线的方程为\\blank{100}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "填空题", "ans": "", @@ -57449,7 +57576,8 @@ "content": "已知$\\triangle ABC$三边所在的直线分别为$4x-y=3$, $x+y=7$, $3x-2y=1$. $\\triangle ABC$的垂心$H$的坐标为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "填空题", "ans": "", @@ -57473,7 +57601,8 @@ "content": "已知$\\triangle ABC$两个顶点的坐标分别为$A(-2,1),B(4,-3)$, $\\triangle ABC$的垂心坐标为$H(0,2)$.\n求$BC$边所在直线的方程.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -57499,7 +57628,8 @@ "content": "三角形$ABC$中，已知$A(1,0),B(1,1),C(5,3)$, 求角$A$的内角平分线所在直线的方程.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -57523,7 +57653,8 @@ "content": "直线$l$过点$(2,3)$, 它的法向量是直线$x-2y=0$的方向向量, 则直线$l$的方程为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "填空题", "ans": "", @@ -57547,7 +57678,8 @@ "content": "若三点$A(3,1)$, $B(-2,b)$, $C(8,11)$同在一条直线上, 则实数$b$等于\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "填空题", "ans": "", @@ -57571,7 +57703,8 @@ "content": "在$x,y$轴上截距分别是$3$, $4$的直线的方程是\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "填空题", "ans": "", @@ -57595,7 +57728,8 @@ "content": "已知直线$l$过点$(3,-1)$, 且与两坐标轴为成一个等腰三角形, 则直线$l$的方程为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "填空题", "ans": "", @@ -57619,7 +57753,8 @@ "content": "若直线$(m+2)x+(m^2-2m-3)y=2m$在$x$轴上的截距是$3$, 则实数$m$的值为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "填空题", "ans": "", @@ -57669,7 +57804,8 @@ "K0705003X" ], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "填空题", "ans": "$4x+3y\\pm 20=0$", @@ -57693,7 +57829,8 @@ "content": "过点$(2,3)$, 且在两条坐标轴上的截距相等的直线方程为\\blank{100}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "填空题", "ans": "", @@ -57717,7 +57854,8 @@ "content": "已知$\\triangle ABC$的两个顶点坐标$A(0,0)$, $B(21,0)$.\\\\ \n(1) 若三角形重心$G$的坐标为$(10,3)$, 求$AC$边所在直线的方程;\\\\ \n(2) 若三角形垂心$H$的坐标为$(15,6)$, 求点$C$的坐标;\\\\ \n(3) 若点$C$的坐标为$(16,12)$, 求三角形内心$I$的坐标.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -57741,7 +57879,8 @@ "content": "在$\\triangle ABC$中, 已知点$A(5,-2)$, $B(7,3)$, 且边$AC$的中点$M$在$y$轴上, 边$BC$的中点$N$在$x$轴上. 求:\\\\ \n(1) 顶点$C$的坐标;\\\\ \n(2) 直线$MN$的方程.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -57765,7 +57904,8 @@ "content": "(利用截距式方程求解)\n已知直线$l$过点$P(3,2)$, 且与$x$正半轴, $y$正半轴分别交于点$A,B$.\\\\ \n(1) 求$\\triangle AOB$面积的最小值及此时$l$的方程($O$为坐标原点);\\\\ \n(2) 求直线$l$在两轴上截距之和的最小值.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -57791,7 +57931,8 @@ "content": "已知直线斜率$k=-2$, 则其倾斜角为\\blank{50}, 一个方向向量为\\blank{50}.\\\\ \n已知直线的一个方向向量为$(1,-3)$, 则其倾斜角为\\blank{50}, 斜率为\\blank{50}.\\\\ \n已知直线的倾斜角为$\\dfrac{\\pi}{6}$, 则其斜率为\\blank{50}, 一个方向向量为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "填空题", "ans": "", @@ -57815,7 +57956,8 @@ "content": "若下列直线的斜率不存在, 则填``不存在''; 若存在, 则写出斜率值.\\\\ \n(1) 直线$2y-1=0$的斜率为\\blank{50};\\\\ \n(2) 直线$2x-1=0$的斜率为\\blank{50};\\\\ \n(3) 设$a,b$为正数, 直线$ax+by-1=0$的斜率为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "填空题", "ans": "", @@ -57839,7 +57981,8 @@ "content": "若$-\\dfrac{\\pi}{2}<\\theta<0$, 则直线$y=x\\cot\\theta$的倾斜角为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "填空题", "ans": "", @@ -57863,7 +58006,8 @@ "content": "已知直线$l$的斜率不大于$\\sqrt{3}$, 则它的倾斜角的取值范围为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "填空题", "ans": "", @@ -57887,7 +58031,8 @@ "content": "若$\\theta\\in \\mathbf{R}$, 则直线$y=x\\sin\\theta+1$的倾斜角的取值范围为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "填空题", "ans": "", @@ -57911,7 +58056,8 @@ "content": "过点$P(2,3)$与$Q(1,5)$的直线$PQ$的倾斜角为\\blank{50}, 点斜式方程为\\blank{80}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "填空题", "ans": "", @@ -57935,7 +58081,8 @@ "content": "已知直线$l$的倾斜角的正弦值为$\\dfrac{3}{5}$, 且过$(1,1)$, 则该直线的斜截式方程为\\blank{100}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "填空题", "ans": "", @@ -57959,7 +58106,8 @@ "content": "已知直线$l$的倾斜角为直线$y=\\sqrt{3}x+1$的倾斜角的一半, 则直线$l$的斜率为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "填空题", "ans": "", @@ -57983,7 +58131,8 @@ "content": "设点$A(2,-3)$, $B(-3,-2)$, 直线$l$过点$P(1,1)$且与线段$AB$相交,\n则$l$的斜率$k$的取值范围为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "填空题", "ans": "", @@ -58007,7 +58156,8 @@ "content": "过原点引直线$l$, 使$l$与连接$A(1,1)$和$B(1,-1)$两点的线段相交,\n则直线$l$倾斜角的取值范围为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "填空题", "ans": "", @@ -58031,7 +58181,8 @@ "content": "设$m$为实常数, 已知两点$M(2m+3,m),N(m-2,1)$, 求直线$MN$的倾斜角.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -58055,7 +58206,8 @@ "content": "若直线$l$的倾斜角是连接$(3,-5)$, $(0,-9)$两点的直线的倾斜角的两倍, 求$l$的斜率.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -58079,7 +58231,8 @@ "content": "[利用点斜式方程求解]\n已知直线$l$过点$P(3,2)$, 且与$x$正半轴, $y$正半轴分别交于点$A,B$.\\\\ \n(1) 求$\\triangle AOB$面积的最小值及此时$l$的方程($O$为坐标原点);\\\\ \n(2) 求直线$l$在两轴上截距之和的最小值.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -58105,7 +58258,8 @@ "content": "如果$pr<0$, $qr<0$, 那么直线$px+qy+r=0$一定不通过第\\blank{40}象限.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "填空题", "ans": "", @@ -58129,7 +58283,8 @@ "content": "直线$x-ay+1=0\\ (a<0)$的斜率为\\blank{50}, 倾斜角为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "填空题", "ans": "", @@ -58153,7 +58308,8 @@ "content": "直线$x-ay+1=0\\ (a\\geq 0)$的倾斜角为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "填空题", "ans": "", @@ -58177,7 +58333,8 @@ "content": "设$a,b$是常数, 过$(a,b)$且平行于直线$2x-y+1=0$的直线方程为\\blank{80}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "填空题", "ans": "", @@ -58203,7 +58360,8 @@ "content": "设$a,b$是常数, 过$(a,b)$且垂直于直线$2x-y+1=0$的直线方程为\\blank{80}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "填空题", "ans": "", @@ -58229,7 +58387,8 @@ "content": "已知矩形$OABC$的顶点$A$的坐标为$(4,3)$, 则直线$AB$的方程为\\blank{100}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "填空题", "ans": "", @@ -58253,7 +58412,8 @@ "content": "若直线与两坐标轴相交, 且被两轴截得的线段中点为$(1,2)$, 则此直线的方程为\\blank{100}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "填空题", "ans": "", @@ -58277,7 +58437,8 @@ "content": "已知$A(-1,0)$, $B(7,2)$, $C(3,8)$是$\\triangle ABC$的三个顶点,\n直线$l$过顶点$C$且平分$\\triangle ABC$的面积, 则$l$的方程为\\blank{100}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "填空题", "ans": "", @@ -58301,7 +58462,8 @@ "content": "直线$l$与直线$y=ax+b(a\\ne 0)$夹角的平分线是直线$y=x$, 则直线$l$的方程是\\blank{100}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "填空题", "ans": "", @@ -58349,7 +58511,8 @@ "content": "[不定项选择]\n已知直线$l:f(x,y)=0$与直线$l$外一点$P(x_0,y_0)$, 那么曲线$f(x,y)-f(x_0,y_0)=0$可能为\\bracket{20}.\n\\twoch{过$P$且与$l$平行的直线}{两条直线}{直线$l$}{与$l$相交的直线}", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "选择题", "ans": "", @@ -58373,7 +58536,8 @@ "content": "已知点$A(x_1,y_1)$, $B(x_2,y_2)$分别在直线$x+y-7=0$和直线$x+y-5=0$上, 求$AB$的中点$M$到原点距离的最小值.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -58397,7 +58561,8 @@ "content": "[利用点斜式方程求解]\n已知$l$经过点$P(1,2)$, 且与两坐标轴围成的三角形面积为$S$.\\\\ \n(1) 当$S=3$时, 满足条件的直线有几条?\\\\ \n(2) 当$S=4$时, 满足条件的直线有几条?\\\\ \n(3) 当$S=5$时, 满足条件的直线有几条?\\\\ \n(4) 设常数$a>0$, 当$S=a$时, 满足条件的直线有几条? (只需写出结果)\\\\ \n注: 观察第(4)小问在几何图形上的直观意义, 再观察第(2)小问与练习7的联系, 以后你可以仅通过心算就可猜出(4)的结果吗?", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -58421,7 +58586,8 @@ "content": "设$t$是常数, 讨论直线$l_1:6x+(t-1)y=8$与直线$l_2:(t+4)x+(t+6)y=16$的位置关系.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -58445,7 +58611,8 @@ "content": "经过两直线$x-2y+4=0$, $x+y-2=0$的交点, 且与直线$3x-4y+5=0$垂直的直线的方程为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "填空题", "ans": "", @@ -58469,7 +58636,8 @@ "content": "若直线$x-2y+4=0$经过直线$x+y-2=0$和$x+ay+8=0$的交点, 则实数$a=$\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "填空题", "ans": "", @@ -58493,7 +58661,8 @@ "content": "若直线$y=kx+k+2$与直线$y=-2x+4$有交点, 且交点在第一象限内, 则实数$k$的取值范围为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "填空题", "ans": "", @@ -58517,7 +58686,8 @@ "content": "若直线$(2m^2+m-3)x+(m^2-m)y=4m-1$与直线$2x-3y=5$互相平行, 则实数$m$的值为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "填空题", "ans": "", @@ -58541,7 +58711,8 @@ "content": "若直线$mx-2y=1$与直线$6x-4y+n=0$重合, 则实数$m,n$的值分别为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "填空题", "ans": "", @@ -58567,7 +58738,8 @@ "content": "已知无论实数$m$取何值, 直线$(2m-1)x+(m+3)y-(m-11)=0$都通过一个定点, 该定点的坐标为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "填空题", "ans": "", @@ -58615,7 +58787,8 @@ "content": "方程$4x^2-y^2+4x+2y=0$表示的曲线是\\bracket{20}.\n\\twoch{一个点}{两条互相平行的直线}{两条互相垂直的直线}{两条相交但不垂直的直线}", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "选择题", "ans": "", @@ -58639,7 +58812,8 @@ "content": "求证: 若三条两两相交的直线$l_i:a_ix+b_iy+c_i=0 (i=1,2,3)$交于同一点, 则\n$\\left|\\begin{array}{ccc}a_1 & b_1& c_1\\\\a_2& b_2&c_2\\\\a_3&b_3&c_3\\end{array}\\right|=0$.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -58665,7 +58839,8 @@ "content": "求证: 若三条两两相交的直线$l_i:a_ix+b_iy+c_i=0 (i=1,2,3)$满足$\\left|\\begin{array}{ccc}a_1 & b_1& c_1\\\\a_2& b_2&c_2\\\\a_3&b_3&c_3\\end{array}\\right|=0$, 则它们交于同一点.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -58691,7 +58866,8 @@ "content": "[选做, 注意逻辑]\n对于某直线$l$上的任意点$(x,y)$, 点$(x+3y,8x-y)$也在该直线上, 求直线$l$的方程.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -58715,7 +58891,8 @@ "content": "直线$x+y=1$与直线$2x+y=0$的夹角为\\blank{50}, 直线$x=1$与直线$2x+y=0$的夹角为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "填空题", "ans": "", @@ -58739,7 +58916,8 @@ "content": "若直线$l_1: ax+(1-a)y=3$与直线$l_2:(a-1)x+(2a+3)y=2$互相垂直, 则实数$a$的值为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "填空题", "ans": "", @@ -58763,7 +58941,8 @@ "content": "若直线$l_1$和$l_2$的斜率是方程$6x^2+x-1=0$的两根, 则$l_1$与$l_2$的夹角为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "填空题", "ans": "", @@ -58787,7 +58966,8 @@ "content": "若直线$l$过原点, 且与直线$y=\\sqrt{3}x+1$夹角为$30^\\circ$, 则直线$l$的方程为\\blank{100}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "填空题", "ans": "", @@ -58811,7 +58991,8 @@ "content": "若等腰直角三角形$ABC$的斜边所在直线的方程是$3x-y+2=0$, 直角顶点是$C(3,-2)$, 则直角边$AC$的方程为\\blank{100}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "填空题", "ans": "", @@ -58883,7 +59064,8 @@ "content": "若直线$l$沿$x$轴的负方向平移$3$个单位, 再沿$y$轴正方向平移$1$个单位后,\n又回到原来的位置, 则直线$l$的斜率为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "填空题", "ans": "", @@ -58907,7 +59089,8 @@ "content": "已知直线$l_1$过点$M$, 将直线$l_1$绕点$M$沿顺时针方向旋转$\\alpha\\in(0,\\dfrac{\\pi}{2})$角, 得到的直线$l_2$的方程为$x+y-2=0$, 将直线$l_2$再绕点$M$顺时针方向旋转$\\dfrac{\\pi}{2}-\\alpha$角, 得到的直线$l_3$的方程为$2x-y-1=0$, 则直线$l_1$的方程为\\blank{100}, $\\alpha=$\\blank{40}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "填空题", "ans": "", @@ -58931,7 +59114,8 @@ "content": "已知$M(-1,-5),N(3,-2)$, 若直线$l$的倾斜角是直线$MN$的倾斜角的一半, 则直线$l$的斜率为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "填空题", "ans": "", @@ -58955,7 +59139,8 @@ "content": "已知等腰三角形的底边过点$P(2,1)$, 两腰所在直线为$x+y-2=0$与$7x-y+4=0$, 求其底边所在直线的方程.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -58979,7 +59164,8 @@ "content": "等腰三角形的一条腰所在直线为${{l}_{1}}:x-2y-2=0$, 底边所在直线为${{l}_{2}}:x+y-1=0$, $(-2,0)$在另一腰上, 求这条腰所在直线${{l}_{3}}$的方程.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -59003,7 +59189,8 @@ "content": "与直线$x-y+\\sqrt{3}=0$关于原点成中心对称的直线方程是\\blank{100}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "填空题", "ans": "", @@ -59027,7 +59214,8 @@ "content": "直线$2x+5y-7=0$关于点$A(1,2)$对称的直线的方程为\\blank{100}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "填空题", "ans": "", @@ -59079,7 +59267,8 @@ "content": "点$(a,b)$关于直线$x+y=0$的对称点是\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "填空题", "ans": "", @@ -59105,7 +59294,8 @@ "content": "点$(2,0)$关于直线$2x-y+1=0$的对称点的坐标是\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "填空题", "ans": "", @@ -59131,7 +59321,8 @@ "content": "点$(a,b)$关于直线$2x-y+1=0$的对称点的坐标是\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "填空题", "ans": "", @@ -59158,7 +59349,8 @@ "content": "直线$y=x+1$关于直线$2x-y+1=0$对称的直线的方程是\\blank{80}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "填空题", "ans": "", @@ -59182,7 +59374,8 @@ "content": "曲线$F(x,y)=0$关于直线$2x-y+1=0$对称的曲线的方程是\\blank{80}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "填空题", "ans": "", @@ -59206,7 +59399,8 @@ "content": "若直线$ax-y+2=0$与直线$3x-y-b=0$关于直线$y=x$对称, 则数对$(a,b)=$\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "填空题", "ans": "", @@ -59230,7 +59424,8 @@ "content": "若点$A(a+2,b+2)$关于直线$4x+3y+11=0$的对称点是$B(b-4,a-b)$, 则$(a,b)=$\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "填空题", "ans": "", @@ -59254,7 +59449,8 @@ "content": "光线通过点$A(2,0)$, 经直线$2x-y+1=0$反射. 若反射线通过点$B(0,-2)$, 则反射光线所在直线的方程为\\blank{80}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "填空题", "ans": "", @@ -59302,7 +59498,8 @@ "content": "已知点$A(2,0)$和$B(4,2)$, 直线$l: x=0$上有一动点$P$.\\\\ \n(1) 求$|PA|+|PB|$的最小值与相应的$P$点的坐标;\\\\ \n(2) $||PA|-|PB||$是否存在最小值? 若存在, 求出最小值与相应的$P$点的坐标, 若不存在, 说明理由;\\\\ \n(3) $|PA|-|PB|$是否存在最大值? 若存在, 求出最大值与相应的$P$点的坐标, 若不存在, 说明理由;\\\\ \n(4)(选做) $|PB|-|PA|$是否存在最大值? 若存在, 求出最大值与相应的$P$点的坐标, 若不存在, 说明理由.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -59326,7 +59523,8 @@ "content": "直线$x+y-4=0$上的点与坐标原点的距离的最小值为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "填空题", "ans": "", @@ -59350,7 +59548,8 @@ "content": "若点$P(a,b)$在直线$x+y+1=0$上, 则$\\sqrt{a^2+b^2-2a-2b+2}$的最小值为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "填空题", "ans": "", @@ -59374,7 +59573,8 @@ "content": "设$p$是实数, 则若点$(1,1)$与点$(p,3)$在直线$2x-3y-1=0$的异侧, 则$p$的取值范围为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "填空题", "ans": "", @@ -59398,7 +59598,8 @@ "content": "已知直线$3x+2y-3=0$与$6x+my+1=0$平行, 则它们之间的距离等于\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "填空题", "ans": "", @@ -59422,7 +59623,8 @@ "content": "已知直线$l$平行于直线$x+y+2=0$, 且这两条直线之间的距离为$3\\sqrt{2}$, 则直线$l$的方程为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "填空题", "ans": "", @@ -59446,7 +59648,8 @@ "content": "已知两平行直线$l_1:3x+4y-10=0$与$l_2:3x+4y-25=0$. 又直线$l$和$l_1$之间的距离与$l$和$l_2$之间的距离之比为$2:3$, 那么直线$l$的方程为\\blank{100}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "填空题", "ans": "", @@ -59470,7 +59673,8 @@ "content": "已知正方形$ABCD$的中心为点$(1,1)$, $AB$边所在直线方程为$x+3y+1=0$, 则$AD$边所在直线的方程是\\blank{100}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "填空题", "ans": "", @@ -59494,7 +59698,8 @@ "content": "到原点和直线$x+3y=2$距离相等且又在直线$x+3y=0$上的点$P$的坐标是\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "填空题", "ans": "", @@ -59518,7 +59723,8 @@ "content": "到直线$2x-y+1=0$与直线$x-2y=2$距离相等的点的轨迹方程为\\blank{100}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "填空题", "ans": "", @@ -59542,7 +59748,8 @@ "content": "已知$P_1(1,0)$与$P_2(7,-8)$两点分别在直线$l$的两侧, 且这两点到直线$l$的距离均为$4$, 求直线$l$的方程.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -59568,7 +59775,8 @@ "content": "不求直线$AB$的方程, 仅用定比分点公式解决如下问题: ``已知点$A(x_1,y_1)$, $B(x_2,y_2)$, 不平行于$AB$的直线$ax+by+c=0$交直线$AB$于$P$点, $\\overrightarrow{AP}=\\lambda\\overrightarrow{PB}$, 求分比$\\lambda$.''", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -59592,7 +59800,8 @@ "content": "已知直线${{l}_{1}}:2x+y+1=0$与直线${{l}_{2}}:x-y=0$的交点为$P$.\\\\ \n(1) 若直线${{l}_{3}}$平行于${{l}_{1}}$且过$(9,9)$, 则${{l}_{3}}$的方程为\\blank{100};\\\\ \n(2) 若直线${{l}_{4}}$过$P$与点$(1,2)$, 则${{l}_{4}}$的方程为\\blank{100}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "填空题", "ans": "", @@ -59616,7 +59825,8 @@ "content": "动直线$l: (k+2)x-2k+(2k+1)y-1=0$过定点\\blank{50}, 定点$P(1,3)$到$l$的距离的取值范围为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "填空题", "ans": "", @@ -59640,7 +59850,8 @@ "content": "已知实数$a,b$满足$2a-3b=1$, 一族直线$l:ax+by-5=0$是否过定点? 若过定点, 求出这个定点的坐标; 若不过定点, 请说明理由.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -59664,7 +59875,8 @@ "content": "已知两直线$a_1x+b_1y+1=0$和$a_2x+b_2y+1=0$的交点是$P(1,2)$, 求过两点$Q(a_1,b_1)$, $R(a_2,b_2)$的直线的方程.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -61352,7 +61564,8 @@ "content": "写出分别满足下列条件的椭圆的标准方程.\\\\ \n(1) 焦点坐标$(6,0),(-6,0)$, 且椭圆过$(0,8)$.\\blank{100}\\\\ \n(2) 焦距为$12$, 且椭圆过$(0,8)$.\\blank{100}\\\\ \n(3) 椭圆过点$(0,-2),(1,0)$.\\blank{100}", "objs": [], "tags": [ - "第七单元" + "第七单元", + "椭圆" ], "genre": "填空题", "ans": "", @@ -61373,7 +61586,8 @@ "content": "椭圆$\\dfrac{x^2}{25}+\\dfrac{y^2}{16}=1$的焦点坐标为\\blank{50}, 离心率为\\blank{50}, 准线方程为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "椭圆" ], "genre": "填空题", "ans": "", @@ -61396,7 +61610,8 @@ "content": "椭圆$\\dfrac{x^2}{16}+\\dfrac{y^2}{25}=1$的焦点坐标为\\blank{50}, 离心率为\\blank{50}, 准线方程为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "椭圆" ], "genre": "填空题", "ans": "", @@ -61419,7 +61634,8 @@ "content": "已知$F_1,F_2$是椭圆$\\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1 \\ (a>b>0)$的两个焦点, $AB$是过$F_1$的弦, 则$\\triangle ABF_2$的周长为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "椭圆" ], "genre": "填空题", "ans": "", @@ -61440,7 +61656,8 @@ "content": "若方程$\\dfrac{x^2}{25-m}+\\dfrac{y^2}{16+m}=1$表示焦点在$y$轴上的椭圆, 则实数$m$的取值范围为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "椭圆" ], "genre": "填空题", "ans": "", @@ -61463,7 +61680,8 @@ "content": "求过点$(-\\dfrac{3}{2},\\dfrac{5}{2})$与$(\\sqrt 3,\\sqrt 5)$的椭圆的标准方程.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "椭圆" ], "genre": "解答题", "ans": "", @@ -61507,7 +61725,8 @@ "content": "[选做]\n平面上有一定直线$l$和$l$外一定点$F$. 求证: 当一个动点$P$到$F$的距离和它到$l$的距离之比是一个小于$1$的常数时, 点$P$的轨迹是椭圆.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "椭圆" ], "genre": "解答题", "ans": "", @@ -61528,7 +61747,8 @@ "content": "若方程$\\dfrac{x^2}{k-5}+\\dfrac{y^2}{3-k}=-1$表示椭圆, 则实数$k$的取值范围为\\blank{80}.\n%\\ans{$(3,4)\\cup (4,5)$}", "objs": [], "tags": [ - "第七单元" + "第七单元", + "椭圆" ], "genre": "填空题", "ans": "", @@ -61549,7 +61769,8 @@ "content": "已知椭圆$mx^2+y^2=9$与椭圆$9x^2+25y^2=100$的焦距相等, 则实数$m=$\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "椭圆" ], "genre": "填空题", "ans": "", @@ -61570,7 +61791,8 @@ "content": "已知$bb>0)$上的任一点, $F_1,F_2$是它的左, 右焦点, 则$|PF_1|\\times |PF_2|$的取值范围为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "椭圆" ], "genre": "填空题", "ans": "", @@ -61717,7 +61944,8 @@ "content": "已知$F_1$是椭圆$5x^2+9y^2=45$的左焦点, $P$是此椭圆上的动点, $A(1,1)$是一定点, 则$|PA|+|PF_1|$的最大值为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "椭圆" ], "genre": "填空题", "ans": "", @@ -61738,7 +61966,8 @@ "content": "已知点$P$在椭圆$\\dfrac{x^2}{100}+\\dfrac{y^2}{36}=1$上, 它到椭圆左焦点$F_1$的距离是\n它到椭圆右焦点$F_2$的距离的$3$倍.\\\\ \n(1) 求$|PF_1|,|PF_2|$;\\\\ \n(2) 求点$P$的坐标.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "椭圆" ], "genre": "解答题", "ans": "", @@ -61759,7 +61988,8 @@ "content": "已知椭圆$\\dfrac{x^2}{2}+y^2=1$,\n直线$l$交椭圆于$A,B$两点, 若线段$AB$的中点坐标为$M(\\dfrac{1}{2},\\dfrac{1}{2})$, 求直线$l$的方程.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "椭圆" ], "genre": "解答题", "ans": "", @@ -61780,7 +62010,8 @@ "content": "已知椭圆$C: \\dfrac{x^2}{9}+y^2=1$, $P$是曲线上的动点, 定点$A$的坐标为$(m,0)$, 其中$m$是实常数.\n求$|PA|$的最小值.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "椭圆" ], "genre": "解答题", "ans": "", @@ -61801,7 +62032,8 @@ "content": "一个焦点把长轴分成长度为$7$和$1$两段的椭圆的标准方程为\\blank{100}.\n%\\ans{$x^2/16+y^2/7=1$or$x^2/7+y^2/16=1$}", "objs": [], "tags": [ - "第七单元" + "第七单元", + "椭圆" ], "genre": "填空题", "ans": "", @@ -61822,7 +62054,8 @@ "content": "已知长轴长与短轴长之比为$2:1$, 一条准线方程为$x+4=0$的椭圆的标准方程为\\blank{100}.\n%\\ans{$\\dfrac{x^2}{12}+\\dfrac{y^2}{3}=1$}", "objs": [], "tags": [ - "第七单元" + "第七单元", + "椭圆" ], "genre": "填空题", "ans": "", @@ -61843,7 +62076,8 @@ "content": "以直线$3x+4y-12=0$和两轴的交点之一作为顶点, 另一交点作为焦点的椭圆的标准方程为\\blank{100}.\n%\\ans{$x^2/25+y^2/9=1$or$x^2/16+y^2/25=1$}", "objs": [], "tags": [ - "第七单元" + "第七单元", + "椭圆" ], "genre": "填空题", "ans": "", @@ -61864,7 +62098,8 @@ "content": "过点$P(3,0)$, 且长轴长是短轴长的三倍的椭圆的标准方程为\\blank{100}.\n%\\ans{$x^2/9+y^2=1$or$x^2/9+y^2/81=1$}", "objs": [], "tags": [ - "第七单元" + "第七单元", + "椭圆" ], "genre": "填空题", "ans": "", @@ -61885,7 +62120,8 @@ "content": "已知$M$为椭圆上一点, $F_1,F_2$是两个焦点, 且$\\angle MF_1F_2=2\\alpha$, $\\angle MF_2F_1=\\alpha$, 则椭圆的离心率为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "椭圆" ], "genre": "填空题", "ans": "", @@ -61906,7 +62142,8 @@ "content": "已知$P$是椭圆$\\dfrac{x^2}{9}+\\dfrac{y^2}{4}=1$上的点, 且$\\angle F_1PF_2=90^\\circ$($F_1,F_2$是该椭圆的两个焦点), 则$\\triangle F_1PF_2$的面积为\\blank{50}, $P$的坐标为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "椭圆" ], "genre": "填空题", "ans": "", @@ -61927,7 +62164,8 @@ "content": "已知$F$是椭圆$b^2x^2+a^2y^2=a^2b^2 \\ (a>b>0)$的一个焦点, $PQ$是过其中心的一条弦, 则$\\triangle PQF$面积的最大值是\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "椭圆" ], "genre": "填空题", "ans": "", @@ -61948,7 +62186,8 @@ "content": "已知直线$l:y=kx+1$, 若不论$k$取何值, $l$总与椭圆$\\dfrac{x^2}{5}+\\dfrac{y^2}{m}=1$总有公共点,\n则常数$m$的取值范围为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "椭圆" ], "genre": "填空题", "ans": "", @@ -61969,7 +62208,8 @@ "content": "以椭圆的两个焦点为直径端点的圆交椭圆于四个点, 若顺次连接这四个点及两个焦点恰好组成一个正六边形, 则椭圆的离心率为\\blank{50}.\n%\\ans{$\\sqrt{3}-1$}", "objs": [], "tags": [ - "第七单元" + "第七单元", + "椭圆" ], "genre": "填空题", "ans": "", @@ -62011,7 +62251,8 @@ "content": "已知椭圆$x^2+2y^2=12$及$x$轴正向上一定点$A$, 过$A$作斜率为$1$的直线, 此直线被椭圆截得的弦长为$\\dfrac{4\\sqrt{14}}{3}$, 求$A$的坐标.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "椭圆" ], "genre": "解答题", "ans": "", @@ -62032,7 +62273,8 @@ "content": "已知$P$是椭圆$\\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1(a>b>0)$上的点, 且$\\angle F_1PF_2=\\theta$($F_1,F_2$是该椭圆的两个焦点), 试用$a,b,\\theta$表示$\\triangle F_1PF_2$的面积.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "椭圆" ], "genre": "解答题", "ans": "", @@ -62053,7 +62295,8 @@ "content": "已知椭圆$\\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1(a>b>0)$与直线$x+2y-2=0$交于$A,B$两点, $|AB|=5$,\n且$AB$中点的坐标为$(m,\\dfrac{1}{2})$, 求此椭圆的方程. (提示: 算法合适的话, 此题不用联立椭圆与直线方程. )", "objs": [], "tags": [ - "第七单元" + "第七单元", + "椭圆" ], "genre": "解答题", "ans": "", @@ -62122,7 +62365,8 @@ "content": "以椭圆的右焦点$F_2$为圆心作圆, 使这个圆通过椭圆的中心, 且交椭圆于$M$点, 若直线$MF_1$($F_1$为左焦点)是圆$F_2$的切线, 则椭圆的离心率为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "椭圆" ], "genre": "填空题", "ans": "", @@ -62143,7 +62387,8 @@ "content": "已知圆柱底面的直径为$2R$, 一个与底面成$30^\\circ$角的平面截这个圆柱, 截得的曲线是椭圆.\n这个椭圆的离心率为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "椭圆" ], "genre": "填空题", "ans": "", @@ -62164,7 +62409,8 @@ "content": "已知椭圆的中心在原点, 长轴在$x$轴上, 直线$x+y=1$被椭圆截得的弦$AB$长为$2\\sqrt{2}$, 且$AB$的中点与椭圆中心连线的斜率为$\\dfrac{\\sqrt{2}}{2}$, 则这个椭圆的方程为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "椭圆" ], "genre": "填空题", "ans": "", @@ -62185,7 +62431,8 @@ "content": "已知点$P$在圆$x^2+(y-4)^2=1$上移动, 点$Q$在椭圆$\\dfrac{x^2}{4}+y^2=1$上移动, 则$|PQ|$的最大值为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "椭圆" ], "genre": "填空题", "ans": "", @@ -62206,7 +62453,8 @@ "content": "已知$\\triangle ABC$的三个顶点均在椭圆$4x^2+5y^2=80$上, 且点$A$是椭圆短轴的下端点, $\\triangle ABC$的重心是椭圆的右焦点, 求直线$BC$的方程.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "椭圆" ], "genre": "解答题", "ans": "", @@ -62227,7 +62475,8 @@ "content": "已知椭圆$\\dfrac{x^2}{2}+y^2=1$.\\\\ \n(1) 求斜率为$2$的平行弦的中点轨迹方程;\\\\ \n(2) 过$A(2,1)$引椭圆的割线, 若截得的弦的中点落在一条二次曲线上, 求这个二次曲线的方程, 并回答(只需给出答案)中点是否能取遍该二次曲线的每一点?", "objs": [], "tags": [ - "第七单元" + "第七单元", + "椭圆" ], "genre": "解答题", "ans": "", @@ -62248,7 +62497,8 @@ "content": "双曲线$\\dfrac{x^2}{4}-y^2=1$的两个焦点的坐标为\\blank{100}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "双曲线" ], "genre": "填空题", "ans": "", @@ -62271,7 +62521,8 @@ "content": "双曲线$y^2-\\dfrac{x^2}{5}=1$的两个焦点的坐标为\\blank{100}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "双曲线" ], "genre": "填空题", "ans": "", @@ -62294,7 +62545,8 @@ "content": "已知双曲线的焦点为$(6,0)$和$(-6,0)$, 且过$(-5,2)$, 则此双曲线方程为\\blank{100}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "双曲线" ], "genre": "填空题", "ans": "", @@ -62315,7 +62567,8 @@ "content": "若方程$x^2\\sin\\theta+y^2\\cos\\theta=1$表示双曲线, 则$\\theta$的取值范围为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "双曲线" ], "genre": "填空题", "ans": "", @@ -62336,7 +62589,8 @@ "content": "已知双曲线$\\dfrac{x^2}{64}-\\dfrac{y^2}{36}=1$上的点$P$到右焦点的距离为$14$, 则$P$到左准线的距离为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "双曲线" ], "genre": "填空题", "ans": "", @@ -62357,7 +62611,8 @@ "content": "已知双曲线$kx^2-2ky^2+1=0$的一个焦点为$(-4,0)$, 则实数$k=$\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "双曲线" ], "genre": "填空题", "ans": "", @@ -62378,7 +62633,8 @@ "content": "已知双曲线的半焦距为$c$, 两准线间的距离为$d$, 且$c=d$, 则双曲线的离心率等于\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "双曲线" ], "genre": "填空题", "ans": "", @@ -62420,7 +62676,9 @@ "content": "已知$P,Q$分别是椭圆$9x^2+4y^2=36$的两个焦点, $M$在双曲线$9x^2-25y^2=225$上, 则$\\triangle PQM$重心的轨迹方程为\\blank{80}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "椭圆", + "双曲线" ], "genre": "填空题", "ans": "", @@ -62504,7 +62762,8 @@ "content": "[选做]\n已知坐标平面内的定点$P(a,b)$. 根据点$P$的不同位置, 讨论是否存在以${{F}_{1}}(3,0),{{F}_{2}}(-3,0)$为焦点的双曲线过点$P$.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "双曲线" ], "genre": "解答题", "ans": "", @@ -62525,7 +62784,8 @@ "content": "双曲线$kx^2-2ky^2=4$的一条准线是$y=1$, 则实数$k$的值等于\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "双曲线" ], "genre": "填空题", "ans": "", @@ -62546,7 +62806,8 @@ "content": "若方程$\\dfrac{x^2}{2-m}+\\dfrac{y^2}{|m|-3}=1$表示双曲线, 则实数$m$的取值范围为\\blank{80}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "双曲线" ], "genre": "填空题", "ans": "", @@ -62567,7 +62828,9 @@ "content": "与两圆$x^2+y^2=1$和$x^2+y^2-8x+7=0$都相切的圆的圆心轨迹是\\bracket{20}.\n\\twoch{两个椭圆}{两条双曲线}{一条双曲线和一条直线}{一个椭圆和一条双曲线}", "objs": [], "tags": [ - "第七单元" + "第七单元", + "椭圆", + "双曲线" ], "genre": "选择题", "ans": "", @@ -62588,7 +62851,8 @@ "content": "已知$E,F$分别是离心率为$\\dfrac{\\sqrt{5}+1}{2}$的双曲线$\\dfrac{x^2}{a^2}-\\dfrac{y^2}{b^2}=1 \\ (a>0,b>0)$的左顶点与右焦点, 再记$M(0,b)$, 则$\\angle EMF$等于\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "双曲线" ], "genre": "填空题", "ans": "", @@ -62609,7 +62873,8 @@ "content": "已知$P$为双曲线$3x^2-5y^2=15$上的一点, $F_1,F_2$为其两个焦点, 且$S_{\\triangle F_1PF_2}=3\\sqrt{3}$, 求$\\angle F_1PF_2$的大小.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "双曲线" ], "genre": "解答题", "ans": "", @@ -62630,7 +62895,8 @@ "content": "已知双曲线$\\dfrac{{{x}^{2}}}{16}-\\dfrac{{{y}^{2}}}{20}=1$的左右焦点分别为${{F}_{1}},{{F}_{2}}$, 设双曲线上有一动点$P$.\\\\ \n(1) 若$|P{{F}_{1}}|=9$, 求$|P{{F}_{2}}|$;\\\\ \n(2) 若$|P{{F}_{1}}|=19$, 求$|P{{F}_{2}}|$;\\\\ \n(3) 设$a$是实常数, 若$P$到定点$A(a,0)$的距离最小值为$10$, 求$a$的值.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "双曲线" ], "genre": "解答题", "ans": "", @@ -62651,7 +62917,9 @@ "content": "已知椭圆$\\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1\\ (a>b>0)$和双曲线$\\dfrac{x^2}{m^2}-\\dfrac{y^2}{n^2}=1 \\ (m,n>0)$有公共的焦点$F_1,F_2$, $P$是两曲线的一个交点.\\\\ \n(1) 证明: $\\angle F_1PF_2=2\\arctan \\dfrac{n}{b}$;\\\\ \n(2) 证明: $\\triangle F_1PF_2$的面积为$bn$.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "椭圆", + "双曲线" ], "genre": "解答题", "ans": "", @@ -62672,7 +62940,8 @@ "content": "双曲线与其共轭双曲线有共同的\\bracket{20}.\n\\fourch{焦点}{准线}{离心率}{渐近线}", "objs": [], "tags": [ - "第七单元" + "第七单元", + "双曲线" ], "genre": "选择题", "ans": "", @@ -62693,7 +62962,8 @@ "content": "中心在原点, 一个焦点为$(3,0)$的等轴双曲线的方程为\\blank{80}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "双曲线" ], "genre": "填空题", "ans": "", @@ -62714,7 +62984,8 @@ "content": "中心在原点, 一个焦点为$(3,0)$, 一条渐近线方程为$2x-3y=0$的双曲线的方程为\\blank{80}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "双曲线" ], "genre": "填空题", "ans": "", @@ -62735,7 +63006,8 @@ "content": "中心在原点, 焦距为$6$, 一条渐近线方程为$2x-3y=0$的双曲线的标准方程为\\blank{80}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "双曲线" ], "genre": "填空题", "ans": "", @@ -62756,7 +63028,8 @@ "content": "已知双曲线$\\dfrac{x^2}{a^2}-\\dfrac{y^2}{b^2}=1$的离心率$e=\\dfrac{5}{4}$, 半虚轴长为$2$, 则该双曲线的方程为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "双曲线" ], "genre": "填空题", "ans": "", @@ -62777,7 +63050,8 @@ "content": "若双曲线的离心率为$2$, 则它的共轭双曲线的离心率为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "双曲线" ], "genre": "填空题", "ans": "", @@ -62798,7 +63072,8 @@ "content": "双曲线$\\dfrac{x^2}{9}-\\dfrac{y^2}{16}=1$的两条渐近线夹角的大小为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "双曲线" ], "genre": "填空题", "ans": "", @@ -62822,7 +63097,8 @@ "content": "[选做]\n已知函数$y=x+\\dfrac{1}{x}$的图像是双曲线, 则该双曲线的离心率为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "双曲线" ], "genre": "填空题", "ans": "", @@ -62843,7 +63119,8 @@ "content": "求过点$(2,-2)$, 且与双曲线$x^2-2y^2=2$有公共渐近线的双曲线的方程为\\blank{80}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "双曲线" ], "genre": "填空题", "ans": "", @@ -62864,7 +63141,8 @@ "content": "若双曲线的虚轴长为$6$, 一条渐近线的方程为$3x-y=0$, 则此双曲线的标准方程为\\blank{80}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "双曲线" ], "genre": "填空题", "ans": "", @@ -62885,7 +63163,8 @@ "content": "双曲线$\\dfrac{x^2}{a^2}-\\dfrac{y^2}{b^2}=1 \\ (a>0,b>0)$的一条准线$l$与一条渐近线交于$P$点, $F$是与$l$相应的焦点.\\\\ \n(1) 求证: 直线$PF$与这条渐近线垂直;\\\\ \n(2) 求$|PF|$.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "双曲线" ], "genre": "解答题", "ans": "", @@ -62906,7 +63185,8 @@ "content": "已知双曲线$\\dfrac{x^2}{9}-\\dfrac{y^2}{16}=1$的焦点分别为$F_1,F_2$, $P$为双曲线上一点, 满足\n$|PF_1|\\cdot|PF_2|=32$. 求证: $PF_1\\perp PF_2$.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "双曲线" ], "genre": "解答题", "ans": "", @@ -62929,7 +63209,9 @@ "content": "与椭圆$x^2+4y^2=64$有共同焦点, 且一条渐近线的方程为$x+\\sqrt{3}y=0$的双曲线的标准方程为\\blank{80}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "椭圆", + "双曲线" ], "genre": "填空题", "ans": "", @@ -62950,7 +63232,8 @@ "content": "已知双曲线的中心在原点, 且一条渐近线方程为$12x-5y=0$, 一条准线方程为$y=\\dfrac{144}{13}$, 则该双曲线的标准方程为\\blank{80}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "双曲线" ], "genre": "填空题", "ans": "", @@ -62971,7 +63254,8 @@ "content": "已知双曲线以两条坐标轴为对称轴, 点$M(\\dfrac{16}{5},\\dfrac{12}{5})$是其准线和渐近线的交点, 则此双曲线的方程为\\blank{80}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "双曲线" ], "genre": "填空题", "ans": "", @@ -63013,7 +63297,8 @@ "content": "过双曲线$\\dfrac{{{x}^{2}}}{3}-{{y}^{2}}=1$的左焦点${{F}_{1}}$作倾斜角为$\\dfrac{\\pi}{3}$的\n弦$AB$, 求三角形${{F}_{2}}AB$的周长及面积.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "双曲线" ], "genre": "解答题", "ans": "", @@ -63034,7 +63319,8 @@ "content": "设$a$是实常数, 已知直线$y=ax+1$与双曲线$3x^2-y^2=1$.\\\\ \n(1) 若直线与双曲线只有一个公共点, 求$a$的值;\\\\ \n(2) 若直线和双曲线的右支有两个公共点, 求$a$的取值范围;\\\\ \n(3) 在(2)的条件下, 设公共点为$A,B$, 是否存在实数$a$使得以线段$AB$为直径的圆经过坐标原点? 若存在, 求$a$ 的值; 若不存在, 说明理由.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "双曲线" ], "genre": "解答题", "ans": "", @@ -63055,7 +63341,8 @@ "content": "已知双曲线$2x^2-y^2=2$, 试问过点$N(1,1)$能否作一直线与双曲线交于$C,D$两点, 且使$N$为$CD$的中点, 这样的直线如果存在, 求出它的方程, 如果不存在, 说明理由.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "双曲线" ], "genre": "解答题", "ans": "", @@ -63076,7 +63363,8 @@ "content": "[选做]\n已知双曲线$2x^2-y^2=2$, 若过点$N(m,n)$能作一直线与双曲线交于$C,D$两点, 且使$N$为$CD$的中点,\n求$(m,n)$所满足的条件.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "双曲线" ], "genre": "解答题", "ans": "", @@ -63118,7 +63406,8 @@ "content": "已知椭圆$\\dfrac{x^2}{25}+\\dfrac{y^2}{9}=1$上三点$A(x_1,y_1)$, $B(4,y_2)$, $C(x_3,y_3)$和焦点$F(4,0)$的距离依次成等差数列.\\\\ \n(1) 求$x_1+x_3$;\\\\ \n(2) 证明: 线段$AC$的垂直平分线过定点, 并求出此定点的坐标.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "椭圆" ], "genre": "解答题", "ans": "", @@ -63139,7 +63428,8 @@ "content": "已知$AB$是双曲线$\\dfrac{x^2}{a^2}-\\dfrac{y^2}{b^2}=1$过左焦点$F_1$的任意一条弦, 以$AB$为直径的圆被左准线截得圆弧$\\overset\\frown{CD}$, 求证: $\\overset\\frown{CD}$所对的圆心角的度数为定值.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "双曲线" ], "genre": "解答题", "ans": "", @@ -63160,7 +63450,8 @@ "content": "已知双曲线$2x^2-y^2=2$.\\\\ \n(1) 求斜率为$2$的平行弦的中点轨迹方程;\\\\ \n(2) 过$A(2,1)$的直线与双曲线交于$P,Q$两点, 线段$PQ$中点$M$落在一条二次曲线上, 求这个二次曲线的方程, 并回答(不需要理由)中点是否能取到该二次曲线上的每一点.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "双曲线" ], "genre": "解答题", "ans": "", @@ -63181,7 +63472,8 @@ "content": "已知直线$l$和双曲线$\\dfrac{x^2}{a^2}-\\dfrac{y^2}{b^2}=1(a>0,b>0)$有两个交点$A,B$. 与该双曲线的渐近线也有两个交点$C,D$. 证明: $|AC|=|BD|$.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "双曲线" ], "genre": "解答题", "ans": "", @@ -63202,7 +63494,8 @@ "content": "若$A$是直线$l$外的一定点, 则过$A$且与$l$相切的圆的圆心轨迹是\\bracket{20}.\n\\fourch{圆}{双曲线一支}{抛物线}{以上都不是}", "objs": [], "tags": [ - "第七单元" + "第七单元", + "抛物线" ], "genre": "选择题", "ans": "", @@ -63248,7 +63541,8 @@ "content": "抛物线$y^2=10x$的焦点到准线的距离是\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "抛物线" ], "genre": "填空题", "ans": "", @@ -63293,7 +63587,8 @@ "content": "在抛物线$y^2=8x$上有一点$P$, 它到焦点的距离为$20$, 则$P$点的坐标为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "抛物线" ], "genre": "填空题", "ans": "", @@ -63314,7 +63609,8 @@ "content": "抛物线$y^2=2x$的焦点弦(过焦点的弦)的端点为$A(x_1,y_1)$与$B(x_2,y_2)$, 且$x_1+x_2=3$, 则$|AB|=$\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "抛物线" ], "genre": "填空题", "ans": "", @@ -63356,7 +63652,8 @@ "content": "过抛物线$y^2=2px$的对称轴上一点$C(p,0)$作一条直线与抛物线交于$A,B$两点,\n若$A$点的纵坐标为$-\\dfrac{p}{2}$, 则$B$点的纵坐标为\\blank{80}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "抛物线" ], "genre": "填空题", "ans": "", @@ -63377,7 +63674,8 @@ "content": "若正三角形的一个顶点在原点, 另两个顶点在抛物线$y^2=2px \\ (p>0)$上, 则这个三角形的面积为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "抛物线" ], "genre": "填空题", "ans": "", @@ -63398,7 +63696,8 @@ "content": "已知$F$是抛物线$y^2=4x$的焦点, $A(3,2)$是一个定点, $P$是抛物线上的动点, 当$|PA|+|PF|$取到最小值时, 点$P$的坐标为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "抛物线" ], "genre": "填空题", "ans": "", @@ -63419,7 +63718,8 @@ "content": "过抛物线$y^2=2px(p>0)$的焦点的一条直线与抛物线相交于两个不同的点, 若两个交点的纵坐标分别为$y_1,y_2$,\n则$y_1y_2$的值为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "抛物线" ], "genre": "填空题", "ans": "", @@ -63442,7 +63742,8 @@ "content": "设过抛物线$y^2=2px(p>0)$的焦点的直线交抛物线于$A,B$两点, 若直线$AB$的倾斜角为$\\theta$,\n求$|AB|$.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "抛物线" ], "genre": "解答题", "ans": "", @@ -63463,7 +63764,8 @@ "content": "抛物线$x^2=-32y$的焦点坐标为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "抛物线" ], "genre": "填空题", "ans": "", @@ -63493,7 +63795,8 @@ "content": "抛物线$y=ax^2 \\ (a\\ne 0)$的准线方程为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "抛物线" ], "genre": "填空题", "ans": "", @@ -63514,7 +63817,8 @@ "content": "抛物线$y^2=16x$上的一点$P$到$x$轴的距离为$12$, 则$P$与焦点$F$间的距离$|PF|=$\\blank{80}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "抛物线" ], "genre": "填空题", "ans": "", @@ -63535,7 +63839,8 @@ "content": "已知抛物线顶点在原点, 焦点在$y$轴上, 又抛物线上一点$(m,-3)$到焦点的距离为$5$, 则此抛物线的方程为\\blank{80}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "抛物线" ], "genre": "填空题", "ans": "", @@ -63556,7 +63861,8 @@ "content": "已知抛物线的顶点在原点, 对称轴与坐标轴重合, 且过点$(-2,3)$, 则抛物线的标准方程为\\blank{80}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "抛物线" ], "genre": "填空题", "ans": "", @@ -63577,7 +63883,8 @@ "content": "抛物线的顶点在原点, 焦点在$x$轴上, 其通径(过焦点, 且与轴垂直的弦)的两端点与顶点连成的三角形面积为$4$, 则此抛物线方程为\\blank{80}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "抛物线" ], "genre": "填空题", "ans": "", @@ -63598,7 +63905,8 @@ "content": "若顶点在原点, 焦点在$x$轴上的抛物线截直线$y=2x+1$所得的弦长为$\\sqrt{15}$, 则此抛物线的方程为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "抛物线" ], "genre": "填空题", "ans": "", @@ -63619,7 +63927,8 @@ "content": "若直线$y=kx-2$交抛物线$y^2=8x$于$A,B$两点, 且$AB$中点的横坐标是$2$, 则$|AB|=$\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "抛物线" ], "genre": "填空题", "ans": "", @@ -63640,7 +63949,8 @@ "content": "已知$AB$是抛物线$y^2=4x$的焦点弦, 其坐标$A(x_1,y_1),B(x_2,y_2)$, 满足$x_1+x_2=6$, 则直线$AB$的斜率是\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "抛物线" ], "genre": "填空题", "ans": "", @@ -63724,7 +64034,8 @@ "content": "抛物线$y^2=-8x$被点$(-1,1)$平分的弦所在直线的方程为\\blank{80}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "抛物线" ], "genre": "填空题", "ans": "", @@ -63745,7 +64056,8 @@ "content": "抛物线$y=2x^2$的一组斜率为$2$的平行弦的中点的轨迹方程是\\blank{80}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "抛物线" ], "genre": "填空题", "ans": "", @@ -63766,7 +64078,8 @@ "content": "已知抛物线$y^2=2x$的弦$AB$过定点$(-2,0)$, 则$AB$中点的轨迹方程是\\blank{80}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "抛物线" ], "genre": "填空题", "ans": "", @@ -63787,7 +64100,8 @@ "content": "过抛物线焦点$F$的直线交此抛物线于$A,B$两点, 弦$AB$的垂直平分线交此曲线的对称轴于$R$. 证明: $|FR|=\\dfrac{1}{2}|AB|$.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "抛物线" ], "genre": "解答题", "ans": "", @@ -63808,7 +64122,8 @@ "content": "已知过抛物线焦点$F$的直线与抛物线相交于$A,B$两点, 点$A,B$在此抛物线准线上的射影分别为$A_1,B_1$. 证明: $\\angle A_1FB_1=90^\\circ$.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "抛物线" ], "genre": "解答题", "ans": "", @@ -63829,7 +64144,8 @@ "content": "已知长度固定的线段$AB$的端点$A,B$在抛物线$y={{x}^{2}}$上移动.\\\\ \n(1) 若$|AB|=2$, 求$AB$的中点$M$到$x$轴的距离的最小值;\\\\ \n(2) (选做)若$|AB|=l(l>0)$, 求$AB$的中点$M$到$x$轴的距离的最小值.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "抛物线" ], "genre": "解答题", "ans": "", @@ -63850,7 +64166,8 @@ "content": "抛物线$y^2=2px \\ (p>0)$的弦$PQ$的中点为$(x_0,y_0)$, 则弦$PQ$所在直线的一个法向量为\\blank{80}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "抛物线" ], "genre": "填空题", "ans": "", @@ -63871,7 +64188,8 @@ "content": "设抛物线$y=ax^2 \\ (a>0)$与直线$y=kx+b$相交于两点, 它们的横坐标为$x_1,x_2$, 而$x_3$是直线与$x$轴交点的横坐标, 那么$x_1,x_2,x_3$的关系是\\bracket{20}.\n\\twoch{$x_3=x_1+x_2$}{$x_3=\\dfrac{1}{x_1}+\\dfrac{1}{x_2}$}{$x_1x_2=x_2x_3+x_3x_1$}{$x_3=\\sqrt{x_1x_2}$}", "objs": [], "tags": [ - "第七单元" + "第七单元", + "抛物线" ], "genre": "选择题", "ans": "", @@ -63892,7 +64210,8 @@ "content": "抛物线$y^2=x$上的点到直线$x-2y+4=0$的距离最小的点是\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "抛物线" ], "genre": "填空题", "ans": "", @@ -63913,7 +64232,8 @@ "content": "若点$P$在抛物线$y^2=x$上, 点$Q$在圆$(x-3)^2+y^2=1$上, 则$|PQ|$的最小值等于\\blank{80}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "抛物线" ], "genre": "填空题", "ans": "", @@ -63934,7 +64254,8 @@ "content": "设$A,B$是抛物线$y^2=2px$上的点, 且满足$\\angle AOB=90^\\circ$($O$是坐标原点). 证明, 直线$AB$过定点, 并求此定点的坐标. (注: 此题可改编成``证明在抛物线的轴上存在一点$P$, 使得过$P$的弦的两端点$AB$总是满足$\\angle AOB=90^\\circ$, 其中$O$是抛物线的顶点'')", "objs": [], "tags": [ - "第七单元" + "第七单元", + "抛物线" ], "genre": "解答题", "ans": "", @@ -63955,7 +64276,8 @@ "content": "已知抛物线$y^2=4x$与椭圆$\\dfrac{x^2}{9}+\\dfrac{y^2}{m}=1$有共同的焦点$F_2$.\\\\ \n(1) 求$m$的值;\\\\ \n(2) 若$P$是两曲线的一个公共点, $F_1$是椭圆的另一个焦点, 且$\\angle PF_1F_2=\\alpha$, $\\angle PF_2F_1=\\beta$, 求$\\cos\\alpha\\cos\\beta$.\\\\ \n(3) 求$\\triangle PF_1F_2$的面积.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "椭圆" ], "genre": "解答题", "ans": "", @@ -63976,7 +64298,8 @@ "content": "已知正方形的一条边$AB$在直线$y=x+4$上, 顶点$C,D$在抛物线$y^2=x$上, 求此正方形的边长.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "抛物线" ], "genre": "解答题", "ans": "", @@ -63997,7 +64320,8 @@ "content": "[选做]\n已知$PQ$是圆$x^2+y^2=1$中的一条垂直于$x$轴的, 不同于直径的定弦. 求证: 所有被$PQ$平分的弦所在的直线都与同一条抛物线有且仅有一个公共点.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "抛物线" ], "genre": "解答题", "ans": "", @@ -64018,7 +64342,8 @@ "content": "抛物线$y^2=-4x$关于直线$x+y-2=0$对称所得曲线的方程是\\blank{80}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "抛物线" ], "genre": "填空题", "ans": "", @@ -64044,7 +64369,8 @@ "content": "抛物线$y^2=-4x$关于直线$x+2y-2=0$对称所得曲线的方程是\\blank{80}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "抛物线" ], "genre": "填空题", "ans": "", @@ -64070,7 +64396,8 @@ "content": "已知双曲线$x^2-\\dfrac{y^2}{3}=1$.\\\\ \n(1) 若双曲线上存在两点关于直线$y=-\\dfrac{1}{3}x+b$对称, 求实数$b$的取值范围;\\\\ \n(2) (选做)若双曲线上存在两点关于直线$y=kx+4$对称, 求实数$k$的取值范围.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "双曲线" ], "genre": "解答题", "ans": "", @@ -64094,7 +64421,8 @@ "content": "已知抛物线$x^2=3y$.\\\\ \n(1) 求该抛物线过点$A(3,3)$的切线的方程;\\\\ \n(2) 求该抛物线过点$B(1,-1)$的切线的方程.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "抛物线" ], "genre": "解答题", "ans": "", @@ -64118,7 +64446,8 @@ "content": "已知双曲线$\\dfrac{y^2}{4}-x^2=1$.\\\\ \n(1) 求该双曲线过点$A(\\sqrt{3},4)$的切线的方程;\\\\ \n(2) 求该双曲线过点$B(1,1)$的切线的方程.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "双曲线" ], "genre": "解答题", "ans": "", @@ -64142,7 +64471,9 @@ "content": "关于过圆锥曲线上一点的切线.\\\\ \n(1) 求过双曲线$\\dfrac{y^2}{a^2}-\\dfrac{x^2}{b^2}=1$上一点$P(x_0,y_0)$的切线方程;\\\\ \n(2) 求过抛物线$x^2=2py(p>0)$上一点$P(x_0,y_0)$的切线方程.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "双曲线", + "抛物线" ], "genre": "解答题", "ans": "", @@ -64166,7 +64497,9 @@ "content": "关于``切点弦''.\\\\ \n(1) 过点$P(x_0,y_0)$引双曲线$\\dfrac{y^2}{a^2}-\\dfrac{x^2}{b^2}=1$的切线, 若有两条切线, 设切点分别为\n$A,B$, 求直线$AB$的方程;\\\\ \n(2) 过点$P(x_0,y_0)$引抛物线$x^2=2py(p>0)$的切线, 若有两条切线, 设切点分别为\n$A,B$, 求直线$AB$的方程.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "双曲线", + "抛物线" ], "genre": "解答题", "ans": "", @@ -64190,7 +64523,9 @@ "content": "关于光学性质.\\\\ \n(1) 已知双曲线$\\dfrac{y^2}{a^2}-\\dfrac{x^2}{b^2}=1$的上下焦点分别为$F_1,F_2$,\n设$P(x_0,y_0)$在双曲线上, 直线$l$为过点$P$的双曲线的切线. 求证: $\\angle F_1PF_2$被直线$l$平分;\\\\ \n(2) 已知抛物线$x^2=2py(p>0)$的焦点为$F$,\n设$P(x_0,y_0)$在抛物线上, 直线$l$为过点$P$的抛物线的切线. 求证: 射线$FP$经过直线$l$反射后, 反射光线与$x$轴垂直.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "双曲线", + "抛物线" ], "genre": "解答题", "ans": "", @@ -64214,7 +64549,8 @@ "content": "焦点为$(-3,5)$, 准线为$y=7$的抛物线的方程是\\blank{80}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "抛物线" ], "genre": "填空题", "ans": "", @@ -64238,7 +64574,8 @@ "content": "抛物线$(x+2)^2=-4(y-1)$的准线方程是\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "抛物线" ], "genre": "填空题", "ans": "", @@ -64267,7 +64604,8 @@ "content": "已知抛物线$y^2=a(x+1)$的准线方程是$x=-3$, 则其焦点坐标为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "抛物线" ], "genre": "填空题", "ans": "", @@ -64291,7 +64629,8 @@ "content": "双曲线$9y^2-x^2-2x-10=0$的渐近线方程是\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "双曲线" ], "genre": "填空题", "ans": "", @@ -64315,7 +64654,8 @@ "content": "以$F_1(0,-1)$, $F_2(0,3)$为两个焦点, 又过点$A(2,1)$的椭圆的方程为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "椭圆" ], "genre": "填空题", "ans": "", @@ -64363,7 +64703,8 @@ "content": "将直线$x-2y+b=0$左移一个单位, 再下移两个单位后, 它与抛物线$y^2=4x$有且仅有一个公共点, 则实数$b=$\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "抛物线" ], "genre": "填空题", "ans": "", @@ -64411,7 +64752,8 @@ "content": "椭圆$\\dfrac{(x-1)^2}{16}+\\dfrac{(y-2)^2}{9}=1$关于点$M(2,-1)$对称的椭圆的方程为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "椭圆" ], "genre": "填空题", "ans": "", @@ -64483,7 +64825,8 @@ "content": "顶点在$(1,2)$, 对称轴平行于坐标轴, 且过点$(4,5)$的抛物线的方程为\\blank{80}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "抛物线" ], "genre": "填空题", "ans": "", @@ -64507,7 +64850,8 @@ "content": "双曲线$x^2-y^2+8x+2y+24=0$的焦点坐标是\\blank{80}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "双曲线" ], "genre": "填空题", "ans": "", @@ -64531,7 +64875,8 @@ "content": "[选做]\n求证:\\\\ \n(1) 曲线$y=\\dfrac{1}{x}$是双曲线;\\\\ \n(2) (选做)曲线$y=2x+\\dfrac{1}{x}$是双曲线.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "双曲线" ], "genre": "解答题", "ans": "", @@ -64899,7 +65244,8 @@ "content": "直线$\\left\\{\\begin{array}{l}x=-2+t\\cos 30^\\circ,\\\\y=3-t\\sin 60^\\circ,\\end{array}\\right.$的倾斜角为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "填空题", "ans": "", @@ -64923,7 +65269,8 @@ "content": "直线$\\left\\{\\begin{array}{l}x=3+at,\\\\y=-1+bt,\\end{array}\\right.$($t$为参数)过定点\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "填空题", "ans": "", @@ -64947,7 +65294,8 @@ "content": "椭圆$\\left\\{\\begin{array}{l}x=4+2\\cos\\theta,\\\\y=1+5\\sin\\theta,\\end{array}\\right.$的焦点坐标为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "椭圆" ], "genre": "填空题", "ans": "", @@ -64971,7 +65319,8 @@ "content": "双曲线$\\left\\{\\begin{array}{l}x=1+\\sqrt{3}\\tan\\theta,\\\\y=1+3\\sec\\theta,\\end{array}\\right.$的两条渐近线的方程为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "双曲线" ], "genre": "填空题", "ans": "", @@ -64995,7 +65344,8 @@ "content": "过点$P(4,-1)$且与直线$l:\\left\\{\\begin{array}{l}x=3+4t,\\\\y=-2+3t,\\end{array}\\right.$平行的直线在$y$轴上的截距为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "填空题", "ans": "", @@ -65019,7 +65369,8 @@ "content": "已知直线$\\left\\{\\begin{array}{l}x=1-3t,\\\\y=2+4t,\\end{array}\\right.$上点$P$到点$(1,2)$的距离为$2$, 则$P$点的坐标为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "填空题", "ans": "", @@ -65163,7 +65514,8 @@ "content": "已知$A,B$分别是椭圆$x^2+4y^2=4$的右顶点与上顶点, $C$是椭圆在第一象限弧上的任意一点, 求四边形$OACB$面积的最大值.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "椭圆" ], "genre": "解答题", "ans": "", @@ -65187,7 +65539,8 @@ "content": "动线段$CD$的一个端点$C$在椭圆$\\dfrac{x^2}{9}+\\dfrac{y^2}{25}=1$上运动, 另一端点在$x$轴上移动. 已知$|CD|=5$, 求$CD$的中点$M$的轨迹方程.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "椭圆" ], "genre": "解答题", "ans": "", @@ -65211,7 +65564,8 @@ "content": "已知直线$l$过点$P(0,3)$, 倾斜角为$\\alpha$, 且与椭圆$\\dfrac{x^2}{9}+\\dfrac{y^2}{4}=1$交\n于$A,B$两点(可重合), 求$|PA|\\cdot|PB|$的最大值.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "椭圆" ], "genre": "解答题", "ans": "", @@ -65235,7 +65589,8 @@ "content": "如图, $AB,CD$是椭圆$\\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1(a>b>0)$的两条相交弦, 交点为$P$. 两弦与椭圆长轴的夹角均为$\\alpha$. 求证: $A,B,C,D$四点共圆.\n\\begin{center}\n\\begin{tikzpicture}[>=latex][scale = 1.5]\n \\draw [->] (-2.5,0) -- (2.5,0) node [below] {$x$};\n \\draw [->] (0,-1.5) -- (0,1.5) node [left] {$y$};\n \\draw (0,0) node [below left] {$O$};\n \\draw (0,0) ellipse (2 and 1);\n \\draw (-1.3271,0.7481) node [above left] {$D$} -- (1.9447,-0.2334) node [below right] {$B$};\n \\draw (-1.7575,-0.4773) node [below left] {$C$} -- (1.6693,0.5508) node [above right] {$A$};\n \\draw (0.5,0.2) node [above] {$P$};\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [ - "第七单元" + "第七单元", + "椭圆" ], "genre": "解答题", "ans": "", @@ -65259,7 +65614,8 @@ "content": "[选做]\n已知$P(1,1)$是椭圆$\\dfrac{x^2}{16}+\\dfrac{y^2}{4}=1$的弦$AB$的一个三等分点,\n求弦$AB$所在直线的方程.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "椭圆" ], "genre": "解答题", "ans": "", @@ -70267,7 +70623,8 @@ "content": "设椭圆$E$的方程为$\\dfrac{x^2}{4}+\\dfrac{y^2}{3}=1$, $F(-1,0)$是椭圆的左焦点, $P$是椭圆$E$上的一个动点, $A(1,1)$是椭圆内一点.\\\\ \n(1) 求$|PA|+2|PF|$的最小值;\\\\ \n(2) 求$|PA|+|PF|$的最小值及最大值.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "椭圆" ], "genre": "解答题", "ans": "", @@ -70288,7 +70645,8 @@ "content": "已知方程$xy=1$表示双曲线, 求它的任意一组焦点和准线.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "双曲线" ], "genre": "解答题", "ans": "", @@ -70309,7 +70667,8 @@ "content": "已知方程$x^2-2xy+y^2-8x-8y=0$表示一条抛物线, 求它的焦点和准线.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "抛物线" ], "genre": "解答题", "ans": "", @@ -70330,7 +70689,8 @@ "content": "已知方程$7x^2-2xy+7y^2=48$表示一个椭圆, 求它的任意一组焦点和准线.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "椭圆" ], "genre": "解答题", "ans": "", @@ -70351,7 +70711,8 @@ "content": "在双曲线$x^2-3y^2=1$上有两个不同的点$A$与$B$.\\\\ \n(1) 若$A,B$同在右支上, 求直线$AB$倾斜角的范围;\\\\ \n(2) 若$A,B$分别在两支上, 求直线$AB$倾斜角的范围.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "双曲线" ], "genre": "解答题", "ans": "", @@ -70372,7 +70733,8 @@ "content": "设椭圆$E$的方程为$x^2+\\dfrac{y^2}{4}=1$. $AB$是椭圆$E$的一条动弦, 其中点记为$M$.\\\\ \n(1) 若$|AB|=2$, 求$M$的纵坐标的最小值; $-\\dfrac{2\\sqrt{3}}{3}$\\\\ \n(2) 若$|AB|=\\dfrac{1}{2}$, 求$M$的纵坐标的最小值. $-\\dfrac{\\sqrt{15}}{2}$", "objs": [], "tags": [ - "第七单元" + "第七单元", + "椭圆" ], "genre": "解答题", "ans": "", @@ -70393,7 +70755,8 @@ "content": "设$AB$是抛物线的一条过焦点$F$的弦, $A'$及$B'$分别是$A$和$B$在准线上的射影. 证明: $\\angle A'FB'=90^\\circ$.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "抛物线" ], "genre": "解答题", "ans": "", @@ -70414,7 +70777,8 @@ "content": "设$AB$是抛物线的一条过焦点的弦.\\\\ \n(1) 证明: 以$AB$为直径的圆与该抛物线的准线相切;\\\\ \n(2) 证明: 从(1)中的圆和准线的切点$T$出发作已知抛物线的两切线, 切点恰为$A$和$B$.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "抛物线" ], "genre": "解答题", "ans": "", @@ -87366,7 +87730,8 @@ "content": "直线$bx+ay=ab$($a<0,\\ b<0$)的倾斜角为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "填空题", "ans": "", @@ -87387,7 +87752,8 @@ "content": "过原点、且倾斜角为直线$y=\\dfrac 12x-3$的倾斜角两倍的直线方程为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "填空题", "ans": "", @@ -87408,7 +87774,8 @@ "content": "$f(x)=a\\sin x-b\\cos x$($ab\\ne 0$)的一条对称轴方程是$x=\\dfrac{\\pi}4$, 则直线$ax-by+c=0$的倾斜角为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "填空题", "ans": "", @@ -87429,7 +87796,8 @@ "content": "若$\\triangle ABC$顶点的坐标分别为$A(2,3)$, $B(-1,4)$, $C(0,-3)$, 则$BC$边上的高所在的直线的方程是\\blank{50}, $BC$边的中线所在的直线的方程是\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "填空题", "ans": "", @@ -87450,7 +87818,8 @@ "content": "(1) 已知直线$l_1:(a+3)x+(2a+5)y-3=0$和$l_2:(1-2a)x+(a-3)y+4=0$, 若$l_1$的方向向量是$l_2$的法向量, 则$a$的值为\\blank{50};\\\\\n(2)若直线$l_1:mx+2y+6=0$和直线$l_2:x+(m-1)y+m^2-1=0$平行, 则实数$m$的值为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "填空题", "ans": "", @@ -87471,7 +87840,8 @@ "content": "已知直线$l:5x+2y+3=0$.\\\\\n(1) 直线$l_1:3x+7y-13=0$与$l$所成的角的大小为\\blank{50};\\\\\n(2) 若$l_2$经过点$P(2,1)$、且与$l$的夹角等于$\\dfrac{\\pi}4$, 则直线$l_2$的方程为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "填空题", "ans": "", @@ -87492,7 +87862,8 @@ "content": "过点$P(1,2)$作直线$l$, 使它到两点$A(2,3)$、$B(4,-5)$的距离相等, 则直线$l$的方程为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "填空题", "ans": "", @@ -87513,7 +87884,8 @@ "content": "(1) 点$P(-2,-1)$关于直线$l:x+2y-2=0$的对称点$Q$的坐标为\\blank{50};\\\\\n(2) 直线$l_1:y=2x+3$关于直线$l:y=x+1$对称的直线$l_2$的方程为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "填空题", "ans": "", @@ -87534,7 +87906,8 @@ "content": "直线$l$过点$M(-1,2)$且与以$A(-2,-3)$、$B(3,0)$为端点的线段(含端点)有公共点.\\\\\n(1) 求直线$l$的倾斜角$\\alpha$的取值范围;\\\\\n(2) 若直线$l$的斜率存在, 求其斜率$k$的取值范围.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -87576,7 +87949,8 @@ "content": "(1) 求过点$P(1,2)$且在两坐标轴上截距相等的直线方程;\\\\\n(2) 求过点$P(1,2)$并且在两坐标轴上的截距的绝对值相等的直线方程;\\\\\n(3) 直线过点$P(1,2)$分别与$x$轴和$y$轴的正半轴交于$A$、$B$两点, 求使$\\triangle OAB$面积最小的直线方程.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -87597,7 +87971,8 @@ "content": "直线$x-y\\cos \\theta +1=0$的倾斜角$\\alpha$的范围是\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "填空题", "ans": "", @@ -87618,7 +87993,8 @@ "content": "写出满足下列条件的直线方程:\\\\\n(1) 过点$(1,-1)$, 且倾斜角为$\\alpha=\\pi-\\arctan\\dfrac 12$:\\blank{50};\\\\\n(2) 过点$(2,3)$与$(-1,-2)$:\\blank{50};\\\\\n(3) 过点$(2,3)$、方向向量为$\\overrightarrow d=(4,7)$的直线方程是\\blank{50};\\\\\n(4) 过点$(2,3)$、法向量$\\overrightarrow{n}=(8,9)$的直线方程是\\blank{50};\\\\\n(5) 已知直线$l$过直线$l_1:3x-5y-10=0$和$l_2:x+y+1=0$的交点, 且平行于$l_3:x+2y-5=0$, 则直线$l$的方程为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "填空题", "ans": "", @@ -87639,7 +88015,8 @@ "content": "已知两直线$l_1:x+m^2y+6=0$与$l_2:(m-2)x+3my+2m=0$. 若$l_1$、$l_2$相交, 则$m$的取值范围为\\blank{50}; 若$l_1$、$l_2$平行, 则$m$的值为\\blank{50}; 若$l_1$、$l_2$重合, 则$m$的值为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "填空题", "ans": "", @@ -87660,7 +88037,8 @@ "content": "(1) 点$P(-1,-1)$到直线$l:2x-3y-11=0$的距离$d$的值是\\blank{50};\\\\\n(2) 直线$x=3$与直线$2x-y+3=0$的夹角是\\blank{50};\\\\\n(3) 直线$l$过点$P(-4,1)$, 且与直线$m:3x-y+1=0$的夹角大小为$\\arccos\\dfrac{3\\sqrt{10}}{10}$, 则$l$的方程是\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "填空题", "ans": "", @@ -87681,7 +88059,8 @@ "content": "(1) 点$P(-2,-1)$关于直线$x+y-2=0$的对称点的坐标是\\blank{50};\\\\\n(2) 直线$l:x+2y-11=0$关于点$(-1,1)$对称的直线方程是\\blank{50};\\\\\n(3) 直线$m:3x-2y-6=0$关于直线$l:2x-3y+1=0$对称的直线方程是\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "填空题", "ans": "", @@ -87702,7 +88081,8 @@ "content": "将直线$l_1:nx+y-n=0$、$l_2:x+ny-n=0$($n\\in \\mathbf{N}^*$)、$x$轴、$y$轴围成的封闭区域的面积记为$S_n$, 则$\\displaystyle\\lim_{n\\to \\infty}S_n=$\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "填空题", "ans": "", @@ -87748,7 +88128,8 @@ "content": "如图, 用$35$个单位正方形拼成一个矩形, 点$P_1$、$P_2$、$P_3$、$P_4$以及四个标记为``\\begin{tikzpicture} \\filldraw ({-0.0625*sqrt(3)},-0.0625) -- ({0.0625*sqrt(3)},-0.0625) -- (0,0.125) -- cycle; \\end{tikzpicture}''的点在正方形的顶点处, 设集合$\\Omega =\\{P_1,P_2,P_3,P_4\\}$, 点$P\\in \\Omega$, 过$P$作直线$l_P$, 使得不在$l_P$上的``\\begin{tikzpicture} \\filldraw ({-0.0625*sqrt(3)},-0.0625) -- ({0.0625*sqrt(3)},-0.0625) -- (0,0.125) -- cycle; \\end{tikzpicture}''的点分布在$l_P$的两侧. 用$D_1(l_P)$和$D_2(l_P)$分别表示$l_P$一侧和另一侧的``\\begin{tikzpicture} \\filldraw ({-0.0625*sqrt(3)},-0.0625) -- ({0.0625*sqrt(3)},-0.0625) -- (0,0.125) -- cycle; \\end{tikzpicture}''的点到$l_P$的距离之和. 若过$P$的直线$l_P$中有且只有一条满足$D_1(l_P)=D_2(l_P)$, 则$\\Omega$中所有这样的$P$为\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, line cap = round, line join = round]\n \\foreach \\i in {0,1,...,7}\n \\draw (\\i,0) -- (\\i,5);\n \\foreach \\i in {0,1,...,5}\n \\draw (0,\\i) -- (7,\\i);\n \\filldraw (0,4) circle (0.05) node [left] {$P_1$};\n \\filldraw (3,2) circle (0.05) node [below left] {$P_2$};\n \\filldraw (4,2) circle (0.05) node [below right] {$P_3$};\n \\filldraw (6,5) circle (0.05) node [above] {$P_4$};\n \\filldraw (1,0) ++ (210:0.1) --++ ({0.1*sqrt(3)},0) --++ (120:{0.1*sqrt(3)}) -- cycle;\n \\filldraw (0,3) ++ (210:0.1) --++ ({0.1*sqrt(3)},0) --++ (120:{0.1*sqrt(3)}) -- cycle;\n \\filldraw (4,4) ++ (210:0.1) --++ ({0.1*sqrt(3)},0) --++ (120:{0.1*sqrt(3)}) -- cycle;\n \\filldraw (7,1) ++ (210:0.1) --++ ({0.1*sqrt(3)},0) --++ (120:{0.1*sqrt(3)}) -- cycle;\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "填空题", "ans": "", @@ -87924,7 +88305,8 @@ "content": "设$f(x,y)=ax+by+c$, 其中$a,b$不全为零. 给定直线$l:f(x,y)=0$及其外一点$P(x_0,y_0)$, 直线$m:f(x,y)-f(x_0,y_0)=0$, 则\\bracket{20}\n\\twoch{点$P$在直线$m$上, 直线$m$与$l$平行}{点$P$在直线$m$上, 直线$m$与$l$不平行}{点$P$在直线$m$外, 直线$m$与$l$平行}{点$P$在直线$m$外, 直线$m$与$l$不平行}", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "选择题", "ans": "", @@ -88155,7 +88537,8 @@ "content": "若点$P(-3,0)$是椭圆$x^2+2y^2-k=0$上的点, 则椭圆的焦点坐标是\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "椭圆" ], "genre": "填空题", "ans": "", @@ -88176,7 +88559,8 @@ "content": "方程$\\dfrac{x^2}{k-5}+\\dfrac{y^2}{3-k}=-1$表示焦点在$y$轴上的椭圆, 则实数$k$的取值范围是\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "椭圆" ], "genre": "填空题", "ans": "", @@ -88197,7 +88581,8 @@ "content": "(1) 焦距是$2\\sqrt 5$, 长轴长是$8$的椭圆的标准方程是\\blank{50};\\\\\n(2) 长轴长是短轴长的2倍, 且经过点$(2,1)$的椭圆的标准方程是\\blank{50};\\\\\n(3) 经过点$A(\\sqrt 3,-2)$、$B(\\sqrt 5,\\dfrac{\\sqrt{30}}3)$的椭圆的方程是\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "椭圆" ], "genre": "填空题", "ans": "", @@ -88218,7 +88603,8 @@ "content": "已知点$P$是椭圆$\\dfrac{x^2}{25}+\\dfrac{y^2}9=1$上一点, $F_1F_2$是焦点, 若$\\angle F_1PF_2=60^\\circ$, 则三角形$F_1PF_2$的面积为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "椭圆" ], "genre": "填空题", "ans": "", @@ -88239,7 +88625,8 @@ "content": "已知椭圆$\\dfrac{x^2}{36}+\\dfrac{y^2}{16}$=1的弦过点$P(3,2)$且被$P$平分, 则此弦所在的直线方程为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "椭圆" ], "genre": "填空题", "ans": "", @@ -88260,7 +88647,8 @@ "content": "椭圆$\\dfrac{x^2}{45}+\\dfrac{y^2}{20}=1$的焦点为$F_1$、$F_2$, 过原点$O$作直线交椭圆于$A$、$B$两点, 若$\\triangle ABF_2$的面积为$20$, 则点$A$的纵坐标为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "椭圆" ], "genre": "填空题", "ans": "", @@ -88302,7 +88690,8 @@ "content": "已知椭圆$\\dfrac{x^2}m+\\dfrac{y^2}6=1$, $F_1,F_2$是它的两个焦点, 若椭圆上存在两个不同的点$P$, 使$\\angle F_1PF_2=90^\\circ$, 则$m=$\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "椭圆" ], "genre": "填空题", "ans": "", @@ -88323,7 +88712,8 @@ "content": "已知椭圆$\\dfrac{y^2}9+{x^2}=1$, 一条不与坐标轴平行的直线$l$与该椭圆交于不同的两点$M$、$N$, 且线段$MN$的中点的横坐标为$-\\dfrac 12$.\\\\\n(1) 求直线$l$的斜率的取值范围;\\\\\n(2) 求直线$l$的倾斜角的取值范围.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "椭圆" ], "genre": "解答题", "ans": "", @@ -88344,7 +88734,8 @@ "content": "已知椭圆$\\dfrac{x^2}2+{y^2}=1$.\\\\\n(1) *过椭圆的左焦点$F$引椭圆的割线, 求截得的弦的中点$P$的轨迹方程;\\\\\n(2) 求斜率为2的平行弦中点$Q$的轨迹方程;\\\\\n(3) 求过点$M(\\dfrac 12,\\dfrac 12)$且被$M$平分的弦所在直线方程.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "椭圆" ], "genre": "解答题", "ans": "", @@ -88365,7 +88756,8 @@ "content": "设椭圆$C:\\dfrac{x^2}2+{y^2}=1$的右焦点为$F$, 过$F$的直线$l$与$C$交于$AB$两点, 点$M$的坐标为$(2,0)$.\\\\\n(1) 当$l$与$x$轴垂直时, 求直线$AM$的方程;\\\\\n(2) 设$O$为坐标原点, 证明: $\\angle OMA=\\angle OMB$.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "椭圆" ], "genre": "解答题", "ans": "", @@ -88386,7 +88778,8 @@ "content": "若椭圆的中心为原点, 焦点在坐标轴上, 焦点到长轴端点的距离分别为$\\sqrt 2-1$与$\\sqrt 2+1$, 则椭圆的方程为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "椭圆" ], "genre": "填空题", "ans": "", @@ -88407,7 +88800,8 @@ "content": "与椭圆$\\dfrac{x^2}9+\\dfrac{y^2}4=1$有相同的焦点, 且经过点$(3,-2)$的椭圆为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "椭圆" ], "genre": "填空题", "ans": "", @@ -88449,7 +88843,8 @@ "content": "椭圆${x^2}+\\dfrac{y^2}4=1$上的点$P(x,y)$到定直线$x+y-6=0$的最远距离是\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "椭圆" ], "genre": "填空题", "ans": "", @@ -88470,7 +88865,8 @@ "content": "记椭圆$\\dfrac{x^2}4+\\dfrac{n{y^2}}{4n+1}=1$围成的区域(含边界)为$\\Omega_n \\ (n=1,2,\\cdots)$, 当点$(x,y)$分别在$\\Omega_1,\\Omega_2,\\cdots$上时, $x+y$的最大值分别是$M_1,M_2,\\cdots$, 则$\\displaystyle\\lim_{n\\to \\infty}M_n=$\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "椭圆" ], "genre": "填空题", "ans": "", @@ -88491,7 +88887,8 @@ "content": "椭圆$\\dfrac{x^2}9+\\dfrac{y^2}4=1$上的动点$P(x,y)$与定点$M(m,0)$($0n>0$)和双曲线$\\dfrac{x^2}{a^2}-\\dfrac{y^2}{b^2}=1$($a>0$, $b>0$)有相同的焦点$F_1,F_2$, 点$P$是椭圆和双曲线的一个交点.\\\\\n(1) 求证: $|PF_1|\\cdot |PF_2|=m^2-a^2$;\\\\\n(2) 求证: $\\triangle PF_1F_2$的面积$S=nb$.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "椭圆", + "双曲线" ], "genre": "解答题", "ans": "", @@ -88792,7 +89201,8 @@ "content": "若双曲线$8mx^2-my^2=8$的一个焦点是$(0,3)$, 则$m=$\\blank{50};", "objs": [], "tags": [ - "第七单元" + "第七单元", + "双曲线" ], "genre": "填空题", "ans": "", @@ -88813,7 +89223,8 @@ "content": "和双曲线$\\dfrac{x^2}9-\\dfrac{y^2}{16}=1$有共同的渐近线, 并且实轴长为$12$的双曲线方程是\\blank{50};", "objs": [], "tags": [ - "第七单元" + "第七单元", + "双曲线" ], "genre": "填空题", "ans": "", @@ -88834,7 +89245,8 @@ "content": "过$P(1,0)$作直线$l$与双曲线${x^2}-\\dfrac{y^2}4=1$只有一个公共点, 则这样的直线共有\\blank{50}条.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "双曲线" ], "genre": "填空题", "ans": "", @@ -88855,7 +89267,8 @@ "content": "已知双曲线$\\dfrac{x^2}{12}-\\dfrac{y^2}4=1$的右焦点为$F$, 若过点$F$的直线$l$与双曲线的右支有且只有一个公共点, 则直线$l$的斜率的取值范围为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "双曲线" ], "genre": "填空题", "ans": "", @@ -88899,7 +89312,8 @@ "content": "求渐近线为$3x\\pm 4y=0$, 焦点为椭圆$\\dfrac{x^2}{10}+\\dfrac{y^2}5=1$的一对顶点的双曲线方程.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "双曲线" ], "genre": "解答题", "ans": "", @@ -88922,7 +89336,8 @@ "content": "已知$F_1F_2$是双曲线${x^2}-\\dfrac{y^2}{b^2}=1$($b>0$)的左、右焦点, 直线$l$过$F_2$且与双曲线交于$AB$两点.\\\\\n(1) 若$l$的倾斜角为$\\dfrac{\\pi}2$, $\\triangle F_1AB$是等边三角形, 求双曲线的渐近线方程;\\\\\n(2) 设$b=\\sqrt 3$, 若$l$的斜率存在, 且$(\\overrightarrow{F_1A}+\\overrightarrow{F_1B})\\cdot \\overrightarrow{AB}=0$, 求$l$的斜率.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "双曲线" ], "genre": "解答题", "ans": "", @@ -88945,7 +89360,8 @@ "content": "设双曲线${x^2}-\\dfrac{y^2}4=1$的右顶点为$A$, 定点$B$的坐标为$(\\dfrac 12,1)$.\\\\\n(1) 是否存在过$B(\\dfrac 12,1)$点且被点$B$平分的双曲线的弦$PQ$, 若存在求出弦$PQ$所在直线方程, 若不存在说明理由;\\\\\n(2) 过点$B$的动直线$l$交双曲线于$P,Q$两点, $M$为线段$PQ$的中点, 求直线$AM$的斜率的取值范围.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "双曲线" ], "genre": "解答题", "ans": "", @@ -88966,7 +89382,8 @@ "content": "已知抛物线的顶点在原点, 焦点在坐标轴上. 分别求适合下列条件的抛物线的标准方程:\\\\\n(1) 过点$(-2,3)$的抛物线为\\blank{50};\\\\\n(2) 准线过点$(2,3)$的抛物线为\\blank{50};\\\\\n(3) 焦点在直线$3x-4y-12=0$上的抛物线为\\blank{50};\\\\\n(4) 焦点在$y$轴上, 抛物线上一点$M(m,-3)$到焦点的距离等于$5$的抛物线为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "抛物线" ], "genre": "填空题", "ans": "", @@ -88987,7 +89404,8 @@ "content": "过点$(2,1)$与抛物线$y=x^2$恰有一个公共点的直线有\\blank{50}条.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "抛物线" ], "genre": "填空题", "ans": "", @@ -89008,7 +89426,8 @@ "content": "抛物线$y=x^2$上到直线$2x-y=4$距离最短的点的坐标为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "抛物线" ], "genre": "填空题", "ans": "", @@ -89029,7 +89448,8 @@ "content": "已知点$A(3,4)$, $F$是抛物线$y^2=8x$的焦点, $M$是抛物线上的动点.当$|MA|+|MF|$最小时, $M$的坐标是\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "抛物线" ], "genre": "填空题", "ans": "", @@ -89050,7 +89470,8 @@ "content": "若$AB$是抛物线$y=x^2$的一条过焦点的弦, 且$|AB|=4$, 则$AB$的中点到直线$y+1=0$的距离是\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "抛物线" ], "genre": "填空题", "ans": "", @@ -89092,7 +89513,8 @@ "content": "已知$F$是抛物线$C:y^2=4x$的焦点, $AB$是抛物线$C$上的两个点, 线段$AB$的中点为$M(2,2)$, 则$\\triangle ABF$的面积等于\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "抛物线" ], "genre": "填空题", "ans": "", @@ -89113,7 +89535,8 @@ "content": "已知$A,B$是抛物线$y^2=2px(p>0)$上的两个点, $O$为坐标原点, 若$|OA|=|OB|$, 且抛物线的焦点恰为$\\triangle AOB$的垂心, 则直线$AB$的方程是\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "抛物线" ], "genre": "填空题", "ans": "", @@ -89155,7 +89578,8 @@ "content": "如图, $M$是抛物线上$y^2=x$上的一点(异于原点), 动弦$ME$、$MF$分别交$x$轴于$AB$两点, 且$MA=MB$.\\\\\n(1) 若$M$为定点, 证明: 直线$EF$的斜率为定值;\\\\\n(2) 若$M$为动点, 且$\\angle EMF={90}^{\\circ}$, 求$\\triangle EMF$的重心$G$的轨迹.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, line cap = round, line join = round, scale = 1.5]\n \\draw [->] (-0.5,0) -- (4,0) node [below] {$x$};\n \\draw [->] (0,-2) -- (0,2) node [left] {$y$};\n \\draw (0,0) node [below left] {$O$}; \n \\draw [domain = -1.8:1.8, samples = 1000] plot (\\x*\\x, \\x);\n \\draw (1.44,1.2) node [above] {$M$};\n \\draw (0.49,-0.7) node [below] {$E$} -- (1.44,1.2) -- (2.89,-1.7) node [below] {$F$} -- cycle;\n \\draw (0.84,0) node [below right] {$A$} (2.04,0) node [above right] {$B$};\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [ - "第七单元" + "第七单元", + "抛物线" ], "genre": "解答题", "ans": "", @@ -89176,7 +89600,8 @@ "content": "求证: 抛物线的准线上任意一点引抛物线的两切线互相垂直并且切点弦过定点.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "抛物线" ], "genre": "解答题", "ans": "", @@ -89197,7 +89622,8 @@ "content": "抛物线$y=-4x^2$的焦点坐标是\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "抛物线" ], "genre": "填空题", "ans": "", @@ -89221,10 +89647,11 @@ }, "003449": { "id": "003449", - "content": "抛物线的焦点在双曲线$\\dfrac{x^2}{25}-\\dfrac{y^2}4=1$上, 则抛物线的标准方程\\blank{50}.\n3.点$A(-4,2)$是抛物线$y^2=-8x$内一点, 抛物线上的点$M$到$A$点的距离与它到焦点的距离之和最小, 则点$M$的坐标是\\blank{50}, 最小距离是\\blank{50}.\n4.设$F$为抛物线$y^2=4x$的焦点, $ABC$为该抛物线上三点.若$\\overrightarrow{FA}+\\overrightarrow{FB}+\\overrightarrow{FC}=\\overrightarrow 0$, 则$|\\overrightarrow{FA}|+|\\overrightarrow{FB}|+|\\overrightarrow{FC}|=$\\blank{50}.\n5.设抛物线$y^2=2x$的焦点为$F$, 过点$M(\\sqrt 3,0)$的直线与抛物线相交于$AB$两点, 与抛物线的准线相交于$C$, $| BF |=2$, 则$\\triangle BCF$与$\\triangle ACF$的面积之比为\\blank{50}.", + "content": "抛物线的焦点在双曲线$\\dfrac{x^2}{25}-\\dfrac{y^2}4=1$上, 则抛物线的标准方程\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "抛物线" ], "genre": "填空题", "ans": "", @@ -89245,7 +89672,8 @@ "content": "设直线$a$与抛物线$\\Omega:y^2=4x$相交于不同的两点$AB$, $O$为坐标原点.\\\\\n(1) 求抛物线$\\Omega$的焦点到准线的距离;\\\\\n(2) 若$\\overrightarrow{OA}\\cdot \\overrightarrow{OB}=0$, 点$Q$在线段$AB$上, 满足$OQ\\perp AB$, 求点$Q$的轨迹.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "抛物线" ], "genre": "解答题", "ans": "", @@ -89266,7 +89694,8 @@ "content": "如图, 已知点$P$是$y$轴左侧(不含$y$轴)一点, 抛物线$C:y^2=4x$上存在不同的两点$A$、$B$满足$PA$、$PB$的中点均在$C$上.\\\\\n(1) 设$AB$中点为$M$, 证明: $PM\\perp y$轴;\\\\\n(2) 若$P$是半椭圆${x^2}+\\dfrac{y^2}4=1x<0$上的动点, 求$\\triangle PAB$面积的取值范围.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, line cap = round, line join = round, scale = 0.7]\n \\draw [->] (-1.5,0) -- (5,0) node [below] {$x$};\n \\draw [->] (0,-5) -- (0,5) node [left] {$y$};\n \\draw (0,0) node [below right] {$O$}; \n \\draw [domain = -4.5:4.5, samples = 1000] plot (\\x*\\x/4, \\x);\n \\draw ({11/4+sqrt(10)/2},{1+sqrt(10)}) node [above] {$A$} coordinate (A);\n \\draw ({11/4-sqrt(10)/2},{1-sqrt(10)}) node [below] {$B$} coordinate (B);\n \\draw ($(A)!0.5!0:(B)$) node [right] {$M$} coordinate (M);\n \\draw (-1,1) node [left] {$P$} coordinate (P);\n \\draw (P) -- (M) (A) -- (P) -- (B) -- cycle;\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [ - "第七单元" + "第七单元", + "抛物线" ], "genre": "解答题", "ans": "", @@ -92927,7 +93356,8 @@ "content": "已知抛物线:$y^2=2px \\ (p>0)$, 若第一象限的$A,B$两点在抛物线上, 焦点为$F$, $|AF|=2$, $|BF|=4$, $|AB|=3$, 则直线$AB$的斜率为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "抛物线" ], "genre": "填空题", "ans": "", @@ -93153,7 +93583,8 @@ "content": "已知椭圆$\\Gamma:\\dfrac{x^2}2+y^2=1$, $F_1, F_2$是其左右焦点, 直线$l$过点$P(m,0) \\ (m<-\\sqrt2)$, 交椭圆$\\Gamma$于$A,B$两点, 且$A,B$都在$x$轴上方, 点$A$在线段$BP$上.\\\\\n(1) 若$B$是上顶点, $|\\overrightarrow{BF_1}|=|\\overrightarrow{PF_1}|$, 求$m$的值;\\\\\n(2) 若$\\overrightarrow{F_1A}\\cdot \\overrightarrow{F_2A}=\\dfrac13$, 且原点$O$到直线$l$的距离为$\\dfrac{4\\sqrt{15}}{15}$, 求直线$l$的方程;\\\\\n(3) 对于任意点$P$, 是否存在唯一直线$l$, 使得$\\overrightarrow{F_1A}\\parallel \\overrightarrow{F_2B}$成立? 若存在, 求出直线$l$的斜率; 若不存在, 请说明理由.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "椭圆" ], "genre": "解答题", "ans": "", @@ -93427,7 +93858,8 @@ "content": "已知椭圆$C:\\dfrac{x^2}4+\\dfrac{y^2}3=1$, 直线$l$经过椭圆右焦点$F$, 交椭圆$C$于$P,Q$两点(点$P$在第二象限), 若$Q$关于$x$轴对称的点为$Q'$, 且满足$PQ\\perp FQ'$, 则直线$l$的方程为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "椭圆" ], "genre": "填空题", "ans": "", @@ -93529,7 +93961,8 @@ "content": "已知直线方程$3x+4y+1=0$的一个参数方程可以是\\bracket{20}.\n\\fourch{$\\begin{cases}x=1+3t, \\\\ y=-1+4t \\end{cases}$}{$\\begin{cases} x=1-4t, \\\\ y=-1-3t \\end{cases}$}{$\\begin{cases} x=1-3t, \\\\ y=-1+4t \\end{cases}$}{$\\begin{cases} x=1+4t, \\\\ y=-1-3t \\end{cases}$}", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "选择题", "ans": "", @@ -93681,7 +94114,8 @@ "content": "双曲线$C_1:\\dfrac{x^2}4-\\dfrac{y^2}{b^2}=1$与圆$C_2:x^2+y^2=4+b^2 \\ (b>0)$交于点$A(x_A,y_A)$(第一象限), 曲线$\\Gamma$由所有在$C_1$或$C_2$上, 且满足$|x|>x_A$的点组成, $C_2$与$x$轴的左、右交点分别记作$F_1,F_2$.\\\\\n(1) 若$x_A=\\sqrt6$, 求$b$的值;\\\\\n(2) 若$b=\\sqrt5$, 点$P$在曲线$\\Gamma$上, 且在第一象限, $|PF_1|=8$, 求$\\angle F_1PF_2$;\\\\\n(3) 点$D(0,\\dfrac{b^2}2+2)$, 过该点的直线斜率为$-\\dfrac b2$的$l$和$\\Gamma$有且只有两个交点, 记作$M,N$, 用$b$表示$\\overrightarrow{OM}\\cdot \\overrightarrow{ON}$, 并求$\\overrightarrow{OM}\\cdot \\overrightarrow{ON}$的取值范围.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "双曲线" ], "genre": "解答题", "ans": "", @@ -93934,7 +94368,8 @@ "content": "过曲线$y^2=4x$的焦点$F$并垂直于$x$轴的直线分别与曲线$y^2=4x$交于$A$、$B$, $A$在$B$的上方, $M$为抛物线上一点, $\\overrightarrow{OM}=\\lambda \\overrightarrow{OA}+(\\lambda-2) \\overrightarrow{OB}$, 则$\\lambda=$\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "抛物线" ], "genre": "填空题", "ans": "", @@ -93978,7 +94413,8 @@ "content": "已知数列$\\{a_n\\}$满足$a_n0)$的焦点为$F_1$、$F_2$, $P$为该双曲线上的一点, 若$|PF_1|=5$, 则$|PF_2|=$\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "双曲线" ], "genre": "填空题", "ans": "", @@ -95044,7 +95485,8 @@ "content": "如图, 用$35$个单位正方形拼成一个矩形, 点$P_1,P_2,P_3,P_4$以及四个标记为``\\begin{tikzpicture}\n\\filldraw ({-0.0625*sqrt(3)},-0.0625) -- ({0.0625*sqrt(3)},-0.0625) -- (0,0.125) -- cycle;\n\\end{tikzpicture}''的点在正方形的顶点处, 设集合$\\Omega=\\{P_1,P_2,P_3,P_4\\}$, 点$P\\in \\Omega$. 过$P$作直线$l_P$, 使得不在$l_P$上的``\\begin{tikzpicture}\n\\filldraw ({-0.0625*sqrt(3)},-0.0625) -- ({0.0625*sqrt(3)},-0.0625) -- (0,0.125) -- cycle;\n\\end{tikzpicture}''的点分布在$l_P$的两侧. 用$D_1(l_P)$和$D_2(l_P)$分别表示$l_P$一侧和另一侧的``\\begin{tikzpicture}\n\\filldraw ({-0.0625*sqrt(3)},-0.0625) -- ({0.0625*sqrt(3)},-0.0625) -- (0,0.125) -- cycle;\n\\end{tikzpicture}''的点到$l_P$的距离之和. 若过$P$的直线$l_P$中有且只有一条满足$D_1(l_P)=D_2(l_P)$, 则$\\Omega$中所有这样的$P$为\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}\n\\foreach \\i in {0,1,2,3,4,5,6,7}{\\draw (\\i,0) -- (\\i,5);};\n\\foreach \\i in {0,1,2,3,4,5}{\\draw (0,\\i) -- (7,\\i);};\n\\filldraw (1,0) ++ ({-0.0625*sqrt(3)},-0.0625) coordinate(P) --++ ({2*0.0625*sqrt(3)},0) --++ ({-0.0625*sqrt(3)},{3*0.0625}) -- (P);\n\\filldraw (7,1) ++ ({-0.0625*sqrt(3)},-0.0625) coordinate(P) --++ ({2*0.0625*sqrt(3)},0) --++ ({-0.0625*sqrt(3)},{3*0.0625}) -- (P);\n\\filldraw (0,3) ++ ({-0.0625*sqrt(3)},-0.0625) coordinate(P) --++ ({2*0.0625*sqrt(3)},0) --++ ({-0.0625*sqrt(3)},{3*0.0625}) -- (P);\n\\filldraw (4,4) ++ ({-0.0625*sqrt(3)},-0.0625) coordinate(P) --++ ({2*0.0625*sqrt(3)},0) --++ ({-0.0625*sqrt(3)},{3*0.0625}) -- (P);\n\\filldraw (0,4) circle (0.05) node [left] {$P_1$};\n\\filldraw (3,2) circle (0.05) node [below left] {$P_2$};\n\\filldraw (4,2) circle (0.05) node [below right] {$P_3$};\n\\filldraw (6,5) circle (0.05) node [above] {$P_4$};\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "填空题", "ans": "", @@ -95139,7 +95581,8 @@ "content": "在平面直角坐标系$xOy$中, 已知椭圆$C_1:\\dfrac{x^2}{36}+\\dfrac{y^2}{4}=1$和$C_2:x^2+\\dfrac{y^2}{9}=1$. $P$为$C_1$上的动点, $Q$为$C_2$上的动点, $w$是$\\overrightarrow{OP}\\cdot \\overrightarrow{OQ}$的最大值. 记$\\Omega=\\{(P,Q)|P\\text{在}C_1\\text{上}, \\ Q\\text{在}C_2\\text{上, 且}\\overrightarrow{OP}\\cdot \\overrightarrow{OQ}=w\\}$, 则$\\Omega$中的元素有\\bracket{15}.\n\\fourch{$2$个}{$4$个}{$8$个}{无穷个}", "objs": [], "tags": [ - "第七单元" + "第七单元", + "椭圆" ], "genre": "选择题", "ans": "", @@ -95231,7 +95674,8 @@ "content": "在平面直角坐标系$xOy$中, 已知椭圆$\\Gamma: \\dfrac{x^2}{4}+y^2=1$, $A$为$\\Gamma$的上顶点, $P$为$\\Gamma$上异于上、下顶点的动点. $M$为$x$正半轴上的动点.\\\\\n(1) 若$P$在第一象限, 且$|OP|=\\sqrt{2}$, 求$P$的坐标;\\\\\n(2) 设$P\\left(\\dfrac{8}{5},\\dfrac{3}{5}\\right)$. 若以$A,P,M$为顶点的三角形是直角三角形, 求$M$的横坐标;\\\\\n(3) 若$|MA|=|MP|$, 直线$AQ$与$\\Gamma$交于另一点$C$, 且$\\overrightarrow{AQ}=2\\overrightarrow{AC}$, $\\overrightarrow{PQ}=4\\overrightarrow{PM}$, 求直线$AQ$的方程.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "椭圆" ], "genre": "解答题", "ans": "", @@ -95519,7 +95963,8 @@ "content": "已知过原点$O$的直线与椭圆$C:\\dfrac{x^2}4+{y^2}=1$交于$A,B$两点, 点$A$到$y$轴的距离$d$满足$d\\in [1,2)$, 点$D$在椭圆$C$上, 且$AD\\perp AB$, 直线$BD$与$x$轴、$y$轴分别交于$M,N$两点.\\\\\n(1) 设直线$BD,AM$的斜率分别为$k_1,k_2$, 求$k_1\\cdot k_2$的取值范围;\\\\\n(2) 求$\\triangle OMN$面积的最大值.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "椭圆" ], "genre": "解答题", "ans": "", @@ -96192,7 +96637,8 @@ "content": "已知椭圆中心在原点, 一个焦点为$F(-2\\sqrt{3},0)$, 且长轴长是短轴长的$2$倍, 则该椭圆的标准方程是\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "椭圆" ], "genre": "填空题", "ans": "", @@ -96320,7 +96766,8 @@ "content": "点$P$在双曲线$\\dfrac{x^2}{a^2}-\\dfrac{y^2}{b^2}=1 \\ (a>0, \\ b>0)$上, $F_1,F_2$是该双曲线的两个焦点, $\\angle F_1PF_2=90^\\circ$, 且$\\triangle F_1PF_2$的三条边长成等差数列, 则$a:b=$\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "双曲线" ], "genre": "填空题", "ans": "", @@ -96528,7 +96975,8 @@ "objs": [], "tags": [ "第三单元", - "第七单元" + "第七单元", + "椭圆" ], "genre": "填空题", "ans": "", @@ -96682,7 +97130,8 @@ "content": "过点$(1,0)$且与直线$x-2y-2=0$的法向量垂直的直线方程是\\bracket{20}.\n\\twoch{$x-2y+1=0$}{$2x+y-2=0$}{$x+2y-1=0$}{$x-2y-1=0$}", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "选择题", "ans": "", @@ -96920,7 +97369,8 @@ "content": "已知双曲线$C_1: \\dfrac{x^2}{a^2}-\\dfrac{y^2}{b^2}=1 \\ (a>0,\\ b>0)$与双曲线$C_2: \\dfrac{x^2}{4}-\\dfrac{y^2}{16}=1$有相同的渐近线, 且$C_1$的右焦点为$F(\\sqrt{5},0)$, 则$a=$\\blank{50}, $b=$\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "双曲线" ], "genre": "填空题", "ans": "", @@ -97281,7 +97731,8 @@ "content": "已知直线$l_1:4x-3y+6=0$和直线$l_2:x+1=0$, 抛物线$y^2=4x$上的动点$P$到直线$l_1$和$l_2$的距离之和的最小值为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "抛物线" ], "genre": "填空题", "ans": "", @@ -97600,7 +98051,8 @@ "content": "某抛物线形拱桥的跨度为$20$米, 拱高是$4$米, 在建桥时, 每隔$4$米需用一根支柱支撑, 其中最高支柱的高度是\\blank{50}米.(答案保留两位小数)", "objs": [], "tags": [ - "第七单元" + "第七单元", + "抛物线" ], "genre": "填空题", "ans": "", @@ -98292,7 +98744,8 @@ "content": "已知$F_1,F_2$为椭圆$\\dfrac{x^2}{25}+\\dfrac{y^2}{16}=1$的左、右焦点, $P$为椭圆上一点, $M$是$F_1P$的中点, $|OM|=3$, 则点$M$到椭圆左焦点的距离为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "椭圆" ], "genre": "填空题", "ans": "", @@ -98533,7 +98986,8 @@ "content": "若抛物线$y^2=2mx$的焦点与双曲线$\\dfrac{x^2}{2}-\\dfrac{y^2}{2}=1$的右焦点重合, 则$m=$\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "抛物线" ], "genre": "填空题", "ans": "", @@ -98991,7 +99445,8 @@ "content": "椭圆$\\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1 \\ (a>b>0)$和圆$x^2+y^2=\\left(\\dfrac b2+c\\right)^2 \\ (c^2=a^2-b^2)$有两个不同的公共点, 则$\\dfrac ca$的值是\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "椭圆" ], "genre": "填空题", "ans": "", @@ -99596,7 +100051,8 @@ "content": "点$P$是双曲线$\\dfrac{x^2}{4}-y^2=1$的右支上一点, $M,N$分别是圆$(x+\\sqrt{5})^2+y^2=1$和圆$(x-\\sqrt{5})^2+y^2=1$上的点, 则$|PM|-|PN|$的最大值是\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "双曲线" ], "genre": "填空题", "ans": "", @@ -99823,7 +100279,8 @@ "content": "已知椭圆$\\dfrac{x^2}{t^2}+\\dfrac{y^2}{5t}=1$的焦距为$2\\sqrt{6}$, 则实数$t=$\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "椭圆" ], "genre": "填空题", "ans": "", @@ -100135,7 +100592,8 @@ "content": "已知椭圆$C:\\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1 \\ (a>b>0)$的一个焦点坐标为$(1,0)$, 且长轴长是短轴长的$\\sqrt{2}$倍.\\\\\n(1) 求椭圆$C$的方程;\\\\\n(2) 设$O$为坐标原点, 椭圆$C$与直线$y=kx+1$相交于两个不同的点$A,B$, 线段$AB$的中点为$P$, 若直线$OP$的斜率为$-1$, 求$\\triangle AOB$的面积.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "椭圆" ], "genre": "解答题", "ans": "", @@ -100333,7 +100791,8 @@ "content": "椭圆两焦点为$F_1(-4,0)$, $F_2(4,0)$, $P$在椭圆上, 若$\\triangle PF_1F_2$的面积的最大值为$12$, 则该椭圆的标准方程为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "椭圆" ], "genre": "填空题", "ans": "", @@ -100596,7 +101055,8 @@ "content": "以抛物线$C:y^2=8x$上的一点$A$为圆心作圆, 若该圆经过抛物线$C$的顶点和焦点, 那么该圆的方程为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "抛物线" ], "genre": "填空题", "ans": "", @@ -100930,7 +101390,8 @@ "content": "从抛物线$y^2=4x$上一点$P$引抛物线的垂线, 垂足为$M$, 且$|PM|=5$, 设抛物线的焦点为$F$, 则$\\triangle MPF$的面积为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "抛物线" ], "genre": "填空题", "ans": "", @@ -101060,7 +101521,8 @@ "content": "双曲线$\\dfrac{x^2}{a^2}-\\dfrac{y^2}{b^2}=1$的左焦点为$F_1$, 顶点为$A_1,A_2$, $P$是该双曲线右支上任意一点, 则分别以线段$PF_1,A_1A_2$为直径的两圆一定\\bracket{20}.\n\\fourch{相交}{内切}{外切}{相离}", "objs": [], "tags": [ - "第七单元" + "第七单元", + "双曲线" ], "genre": "选择题", "ans": "", @@ -101081,7 +101543,8 @@ "content": "方程$x^2+\\sqrt{2}x-1=0$的解可视为函数$y=x+\\sqrt{2}$的图像与函数$y=\\dfrac 1x$的图像交点的横坐标, 若$x^4+ax-4=0$的各个实根$x_1,x_2,\\cdots,x_k \\ (k\\le 4)$所对应的点$\\left(x_i,\\dfrac{4}{x_i}\\right) \\ (i=1,2,\\cdots,k)$均在直线$y=x$的同侧, 则实数$a$的取值范围是\\bracket{20}.\n\\fourch{$(-\\infty,-6)$}{$(6,+\\infty)$}{$[-6,6]$}{$(-\\infty,-6)\\cup (6,+\\infty)$}", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "选择题", "ans": "", @@ -101360,7 +101823,8 @@ "content": "$F$为双曲线$C: \\dfrac{x^2}{64}-\\dfrac{y^2}{16}=1$的左焦点, 双曲线$C$上的点$P_i$与$P_{7-i} \\ (i=1,2,3)$关于$y$轴对称, 且$P_1,P_2,P_3$在双曲线的右支上, 则$|P_1F|+|P_2F|+|P_3F|-|P_4F|-|P_5F|-|P_6F|$的值是\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "双曲线" ], "genre": "填空题", "ans": "", @@ -102072,7 +102536,7 @@ "content": "对于两条不相交的空间直线$a$和$b$, 一定存在平面$\\alpha$, 使得\\bracket{20}.\n\\twoch{直线$a,b$均在平面$\\alpha$内}{直线$a$在平面$\\alpha$内, $b$与平面$\\alpha$平行}{直线$a,b$都垂直于平面$\\alpha$}{直线$a$在平面$\\alpha$内, $b$与平面$\\alpha$垂直}", "objs": [], "tags": [ - "第七单元" + "第六单元" ], "genre": "选择题", "ans": "", @@ -103630,7 +104094,8 @@ "content": "若直线$l$的参数方程为$\\begin{cases} x=4-4t, \\\\ y=-2+3t, \\end{cases}$ $t\\in \\mathbf{R}$, 则直线$l$在$y$轴上的截距是\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "填空题", "ans": "$1$", @@ -103730,7 +104195,8 @@ "content": "焦点在$y$轴上, 焦距为$6$, 且经过点$(0,\\sqrt 5)$的双曲线的标准方程为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "双曲线" ], "genre": "填空题", "ans": "$\\dfrac{y^2}{5}-\\dfrac{x^2}{4}=1$", @@ -103753,7 +104219,8 @@ "content": "已知抛物线型拱桥的顶点距水面$2$米时, 量得水面宽为$8$米, 当水面下降$1$米后, 水面的宽为\\blank{50}米.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "抛物线" ], "genre": "填空题", "ans": "$4\\sqrt{6}$", @@ -103975,7 +104442,8 @@ ], "tags": [ "第二单元", - "第七单元" + "第七单元", + "直线" ], "genre": "选择题", "ans": "A", @@ -104068,7 +104536,10 @@ "content": "(1) 设椭圆$C_1:\\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1$与双曲线$C_2:9{x^2}-\\dfrac{9y^2}8=1$有相同的焦点$F_1$、$F_2$, $M$是椭圆$C_1$与双曲线$C_2$的公共点, 且$\\triangle MF_1F_2$的周长为$6$, 求椭圆$C_1$的方程;\\\\\n我们把具有公共焦点、公共对称轴的两段圆锥曲线弧合成的封闭曲线称为``盾圆''.\\\\\n(2) 如图, 已知``盾圆$D$''的方程为$y^2=\\begin{cases}\n4x, & 0\\le x\\le 3,\\\\ -12(x-4), & 3=latex]\n \\draw [->] (-0.5,0) -- (6,0) node [below] {$x$};\n \\draw [->] (0,-4) -- (0,4) node [left] {$y$};\n \\draw (0,0) node [below left] {$O$};\n \\draw [domain = {-2*sqrt(3)}:{2*sqrt(3)}, samples = 200] plot ({\\x*\\x/4},\\x);\n \\draw [domain = {-2*sqrt(3)}:{2*sqrt(3)}, samples = 200] plot ({\\x*\\x/(-12)+4},\\x);\n \\draw [dashed] (3,-4) -- (3,4);\n \\draw (3,0) node [below left] {$3$};\n \\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [ - "第七单元" + "第七单元", + "椭圆", + "双曲线", + "抛物线" ], "genre": "解答题", "ans": "(1) $\\dfrac{x^2}4+\\dfrac{y^2}3=1$; (2) 定值为$4$; (3) 面积的最大值为$\\sqrt{3}$.", @@ -104222,7 +104693,8 @@ "content": "已知双曲线$\\dfrac{x^2}{a^2}-\\dfrac{y^2}{81}=1$($a>0$)的一条渐近线方程为$y=3x$, 则$a=$\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "双曲线" ], "genre": "填空题", "ans": "", @@ -104673,7 +105145,8 @@ "content": "已知椭圆$C:\\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1$($a>b>0$), 定义椭圆$C$上的点$M(x_0,y_0)$的``伴随点''为$N(\\dfrac{x_0}a,\\dfrac{y_0}b)$.\\\\\n(1) 求椭圆$C$上的点$M$的``伴随点''$N$的轨迹方程;\\\\\n(2) 如果椭圆$C$上的点$(1,\\dfrac 32)$的``伴随点''为$(\\dfrac 12,\\dfrac 3{2b})$, 对于椭圆$C$上的任意点$M$及它的``伴随点''$N$, 求$\\overrightarrow{OM}\\cdot \\overrightarrow{ON}$的取值范围;\\\\\n(3) 当$a=2$, $b=\\sqrt 3$时, 直线$l$交椭圆$C$于$A$, $B$两点, 若点$A$, $B$的``伴随点''分别是$P$, $Q$, 且以$PQ$为直径的圆经过坐标原点$O$, 求$\\triangle OAB$的面积.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "椭圆" ], "genre": "解答题", "ans": "", @@ -105092,7 +105565,8 @@ "tags": [ "第一单元", "第三单元", - "第七单元" + "第七单元", + "直线" ], "genre": "选择题", "ans": "", @@ -105226,7 +105700,8 @@ "content": "已知椭圆$\\Omega :\\dfrac{x^2}4+\\dfrac{y^2}3=1$的左右两焦点分别为$F_1,F_2$.\\\\\n(1) 若矩形$ABCD$的边$AB$在$y$轴上, 点$C,D$均在$\\Omega$上, 求该矩形绕$y$轴旋转一周所得圆柱侧面积$S$的取值范围;\\\\\n(2) 设斜率为$k$的直线$l$与$\\Omega$交于$P,Q$两点, 线段$PQ$的中点为$M(1,m)$($m>0$), 求证: $k<-\\dfrac 12$;\\\\\n(3) 过$\\Omega$上一动点$E(x_0,y_0)$作直线$l:\\dfrac{x_0x}4+\\dfrac{y_0y}3=1$, 其中$y_0\\ne 0$, 过$E$作直线$l$的垂线交$x$轴于点$R$. 问是否存在实数$\\lambda$, 使得$|EF_1|\\cdot |RF_2|=\\lambda |EF_2|\\cdot |RF_1|$恒成立? 若存在, 求出$\\lambda$的值; 若不存在, 说明理由.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "椭圆" ], "genre": "解答题", "ans": "", @@ -105483,7 +105958,8 @@ "content": "设椭圆$\\Gamma:\\dfrac{x^2}{a^2}+y^2=1$($a>1$)的左顶点为$A$, 过点$A$的直线$l$与$\\Gamma$相交于另一点$B$, 与$y$轴相交于点$C$. 若$|OA|=|OC|$, $|AB|=|AC|$, 则$a=$\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "椭圆" ], "genre": "填空题", "ans": "", @@ -105797,7 +106273,8 @@ "content": "已知常数$p>0$, 抛物线$\\Gamma:y^2=2px$的焦点为$F$.\\\\\n(1) 若直线$x=2$被$\\Gamma$截得的弦长为$4$, 求$p$的值;\\\\\n(2) 设$E$为点$F$关于原点$O$的对称点, $P$为$\\Gamma$上的动点, 求$\\dfrac{|PE|}{|PF|}$的取值范围;\\\\\n(3) 设$p=2$. 两条互相垂直的直线$l_1,l_2$均过点$F$, $l_1$与$\\Gamma$相交于$A,B$两点, $l_2$与$\\Gamma$相交于$C,D$两点. 若$AC\\perp BC$, 求四边形$ACBD$的面积.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "抛物线" ], "genre": "解答题", "ans": "", @@ -105933,7 +106410,8 @@ "content": "直线$\\begin{cases} x=-2-\\sqrt 2t, \\\\ y=3+\\sqrt 2t \\end{cases}$($t$为参数)对应的普通方程是\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "填空题", "ans": "", @@ -106093,7 +106571,8 @@ "content": "已知椭圆$x^2+\\dfrac{y^2}{b^2}=1$($00$, $b>0$)的实轴与虚轴长度相等, 则$C$的渐近线方程是\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "双曲线" ], "genre": "填空题", "ans": "", @@ -107259,7 +107743,8 @@ "content": "抛物线$y=4x^2$的准线方程是\\bracket{20}.\n\\fourch{$x=-2$}{$x=-1$}{$y=-\\dfrac 18$}{$y=-\\dfrac 1{16}$}", "objs": [], "tags": [ - "第七单元" + "第七单元", + "抛物线" ], "genre": "选择题", "ans": "", @@ -107424,7 +107909,8 @@ "content": "设椭圆$\\Gamma$: $\\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1$($a>b>0$)的右焦点为$F(1,0)$, 短轴的一个端点$B$到$F$的距离等于焦距.\n(1) 求椭圆$\\Gamma$的标准方程;\\\\\n(2) 设$C$、$D$是四条直线$x=\\pm a$, $y=\\pm b$所围成的矩形在第一、第二象限的两个顶点, $P$是椭圆$\\Gamma$上任意一点, 若$\\overrightarrow{OP}=m\\overrightarrow{OC}+n\\overrightarrow{OD}$, 求证: $m^2+n^2$为定值;\\\\\n(3) 过点$F$的直线$l$与椭圆$\\Gamma$交于不同的两点$M$、$N$, 且满足$\\triangle BFM$与$\\triangle BFN$的面积的比值为$2$, 求直线$l$的方程.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "椭圆" ], "genre": "解答题", "ans": "", @@ -107632,7 +108118,8 @@ "content": "平面上整点(横、纵坐标都为整数的点)到直线$y=\\dfrac 53x+\\dfrac 45$的距离的最小值是\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "填空题", "ans": "", @@ -107828,7 +108315,10 @@ "objs": [], "tags": [ "第六单元", - "第七单元" + "第七单元", + "椭圆", + "双曲线", + "抛物线" ], "genre": "选择题", "ans": "", @@ -107936,7 +108426,8 @@ "content": "设抛物线$y^2=4px\\ (p>0)$的准线与$x$轴的交点为$M$, 过$M$作直线$l$交抛物线于$A$、$B$两点.\\\\\n(1) 求线段$AB$中点的轨迹方程;\\\\\n(2) 若线段$AB$的垂直平分线交对称轴于$N(x_0,0)$, 求$x_0$的取值范围;\\\\\n(3) 若直线$l$的斜率依次取$p, p^2,p^3,\\cdots ,p^n,\\cdots$时, 线段$AB$的垂直平分线与对称轴的交点依次为$N_1, N_2, N_3,\\cdots ,N_n, \\cdots$, 当$0b>0$)经过圆$N:x^2+(y+1)^2=4$与$x$轴的两个交点和与$y$轴正半轴的交点.\\\\\n\\begin{center}\n \\begin{tikzpicture}[>=latex]\n \\draw [->] (-2,0) -- (2,0) node [below] {$x$};\n \\draw [->] (0,-3) -- (0,2) node [left] {$y$};\n \\draw (0,0) node [below left] {$O$};\n \\draw (0,-1) node [below left] {$N$} circle (2) (0,0) ellipse ({sqrt(3)} and 1);\n \\draw ({sqrt(3)*cos(60)},{sin(60)}) node [above] {$P$} -- ({2*cos(-100)},{-1+2*sin(-100)}) node [below] {$Q$};\n \\filldraw (0.25,0.5) circle (0.03) node [below] {$E$};\n \\draw (-1.171,0.7368) node [left] {$A$} -- (-1.0314,0.7136) node [below] {$C$} -- (1.5314,0.2864) node [below] {$D$} -- (1.671,0.2632) node [right] {$B$};\n \\end{tikzpicture}\n\\end{center}\n(1) 求椭圆$M$的方程;\\\\\n(2) 若点$P$为椭圆$M$上的动点, 点$Q$为圆$N$上的动点, 求线段$PQ$长的最大值;\\\\\n(3) 若不平行于坐标轴的直线$L$交椭圆$M$于$A$、$B$两点, 交圆$N$于$C$、$D$两点, 且满足$\\overrightarrow{AC}=\\overrightarrow{DB}$, 求证: 线段$AB$的中点$E$在定直线上.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "椭圆" ], "genre": "解答题", "ans": "", @@ -109855,7 +110350,8 @@ "content": "平面上三条直线$x-2y+1=0$, $x-1=0$, $x+ky=0$, 如果这三条直线将平面划分为六个部分, 则实数$k$的取值组成的集合$A=$\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "填空题", "ans": "", @@ -109880,7 +110376,8 @@ "content": "已知双曲线$C: \\dfrac{x^2}9-\\dfrac{y^2}8=1$, 左、右焦点分别为$F_1$、$F_2$, 过点$F_2$作一直线与双曲线$C$的右支交于$P$、$Q$两点, 使得$\\angle F_1PQ=90^\\circ$, 则$\\triangle F_1PQ$的内切圆的半径$r=$\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "双曲线" ], "genre": "填空题", "ans": "", @@ -109951,7 +110448,8 @@ "content": "椭圆的参数方程为$\\begin{cases} x=5\\cos \\theta, \\\\ y=3\\sin \\theta \\end{cases}$($\\theta$ 为参数), 则它的两个焦点坐标是\\bracket{20}.\n\\fourch{$(\\pm 4,0)$}{$(0,\\pm 4)$}{$(\\pm 5,0)$}{$(0,\\pm 3)$}", "objs": [], "tags": [ - "第七单元" + "第七单元", + "椭圆" ], "genre": "选择题", "ans": "", @@ -110093,7 +110591,8 @@ "content": "已知椭圆$C:\\dfrac{x^2}2+y^2=1$的左、右焦点分别为$F_1$、$F_2$.\\\\\n(1) 点$P$在椭圆$C$上运动(点$P$不在$x$轴上), 设$F_2$关于$\\angle F_1PF_2$的外角平分线所在直线的对称点为$Q$, 求$Q$的轨迹方程;\\\\\n(2) 设$M$、$N$分别是曲线$C$上的两个不同点, 且点$M$在第一象限, 点$N$在第三象限, 若$\\overrightarrow{OM}+2\\overrightarrow{ON}=2\\overrightarrow{OF_1}$, $O$为坐标原点, 求直线$MN$的斜率;\\\\\n(3) 过点$S(0,-\\dfrac 13)$的动直线$l$交曲线$C$于$A$、$B$两点, 在$y$轴上是否存在定点$T$, 使以$AB$为直径的圆恒过这个点? 若存在, 求出点$T$的坐标; 若不存在, 请说明理由.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "椭圆" ], "genre": "解答题", "ans": "", @@ -110680,7 +111179,8 @@ "content": "双曲线$\\Gamma$: $x^2-\\dfrac{y^2}{b^2}=1$($b>0$).\\\\\n(1) 若$\\Gamma$的一条渐近线方程为$y=2x$, 求$\\Gamma$的方程;\\\\\n(2) 设$F_1$、$F_2$是$\\Gamma$的两个焦点, $P$为$\\Gamma$上一点, 且$PF_1\\perp PF_2$, $\\triangle PF_1F_2$的面积为$9$, 求$b$的值;\\\\\n(3) 已知斜率为$2$的直线与$\\Gamma$交于$A$、$B$两点, 点$M$是线段$AB$的中点, 设点$M$的横坐标的集合为$\\Omega$. 若$\\{x|x=2n,\\ n\\in \\mathbf{N}^* \\}\\subseteq \\Omega$, 求正数$b$的取值范围.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "双曲线" ], "genre": "解答题", "ans": "", @@ -111189,7 +111689,8 @@ "content": "设抛物线$\\Gamma$的方程为$y^2=2px$, 其中常数$p>0$. $F$是抛物线$\\Gamma$的焦点.\\\\\n(1) 若直线$x=3$被抛物线$\\Gamma$所截得的弦长为$6$, 求$p$的值;\\\\\n(2) 设$A$是点$F$关于顶点$O$的对称点. $P$是抛物线$\\Gamma$上的动点, 求$\\dfrac{|PA|}{|PF|}$的最大值;\\\\\n(3) 设$p=2$, $l_1,l_2$是两条互相垂直, 且均经过点$F$的直线. $l_1$与抛物线$\\Gamma$交于点$A$、$B$, $l_2$与抛物线交于点$C$、$D$. 若点$G$满足$4\\overrightarrow{FG}=\\overrightarrow{FA}+\\overrightarrow{FB}+\\overrightarrow{FC}+\\overrightarrow{FD}$, 求点$G$的轨迹方程.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "抛物线" ], "genre": "解答题", "ans": "", @@ -111463,7 +111964,8 @@ "content": "设椭圆$\\Gamma:\\dfrac{x^2}{a^2}+{y^2}=1$($a>1$)的左顶点为$A$, 过点$A$的直线$l$与$\\Gamma$相交于另一点$B$, 与$y$轴相交于点$C$. 若$|OA|=|OC|$, $|AB|=|BC|$, 则$a=$\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "椭圆" ], "genre": "填空题", "ans": "", @@ -111769,7 +112271,8 @@ "content": "在平面直角坐标系$xOy$中, 抛物线$\\Gamma:y^2=4x$, 点$C(1,0)$. $A,B$为$\\Gamma$上的两点, $A$在第一象限, 满足$\\overrightarrow{OA}\\cdot \\overrightarrow{OB}=-4$.\\\\\n(1) 求证: 直线$AB$过定点, 并求定点坐标;\\\\\n(2) 设$P$为$\\Gamma$上的动点, 求$\\dfrac{|OP|}{|CP|}$的取值范围;\\\\\n(3) 记$\\triangle AOB$的面积为$S_1$, $\\triangle BOC$的面积为$S_2$, 求$S_1+S_2$的最小值.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "抛物线" ], "genre": "解答题", "ans": "", @@ -114810,7 +115313,8 @@ "content": "已知直线$l:x=t$($0=latex]\n \\draw [->] (-2.5,0) -- (2.5,0) node [below] {$x$};\n \\draw [->] (0,-2) -- (0,2) node [left] {$y$};\n \\draw (0,0) node [below left] {$O$};\n \\draw [name path = ellipse] (0,0) ellipse (2 and {sqrt(2)});\n \\draw [name path = line] (0.8,-1.5) -- (0.8,1.5);\n \\draw [name intersections = {of = line and ellipse, by = {A,B}}];\n \\filldraw (A) circle (0.03) node [above right] {$A$};\n \\filldraw (B) circle (0.03) node [below right] {$B$};\n \\end{tikzpicture}\n\\end{center}\n(1) 记$F_1,F_2$是椭圆$\\Gamma$的左右焦点, 若直线$AB$过$F_2$, 当$M$到$F_1$的距离与到直线$AB$的距离相等时, 求点$M$的横坐标;\\\\\n(2) 若点$M,A$关于$y$轴对称, 当$\\triangle MAB$的面积最大时, 求直线$MB$的方程;\\\\\n(3) 设直线$MA$和$MB$与$x$轴分别交于$PQ$, 证明: $|OP|\\cdot |OQ|$为定值.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "椭圆" ], "genre": "解答题", "ans": "", @@ -115022,7 +115526,8 @@ "content": "在平面直角坐标系$xOy$中, 已知抛物线$y^2=4x$上一点$P$到焦点的距离为$3$, 则点$P$的横坐标是\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "抛物线" ], "genre": "填空题", "ans": "", @@ -115200,7 +115705,8 @@ ], "tags": [ "第一单元", - "第七单元" + "第七单元", + "直线" ], "genre": "选择题", "ans": "", @@ -115523,7 +116029,8 @@ "content": "抛物线$x^2=-4y$的准线方程为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "抛物线" ], "genre": "填空题", "ans": "", @@ -115770,7 +116277,9 @@ "content": "以抛物线$y^2=4x$的焦点为右焦点, 且长轴为$4$的椭圆的标准方程为\\bracket{20}.\n\\fourch{$\\dfrac{x^2}{16}+\\dfrac{y^2}{15}=1$}{$\\dfrac{x^2}{16}+\\dfrac{y^2}4=1$}{$\\dfrac{x^2}4+\\dfrac{y^2}3=1$}{$\\dfrac{x^2}4+{y^2}=1$}", "objs": [], "tags": [ - "第七单元" + "第七单元", + "椭圆", + "抛物线" ], "genre": "选择题", "ans": "", @@ -115890,7 +116399,8 @@ "content": "已知椭圆$C_1:\\dfrac{x^2}4+{y^2}=1$, $F_1$、$F_2$为$C_1$的左、右焦点.\\\\\n(1) 求椭圆$C_1$的焦距;\\\\\n(2) 点$Q(\\sqrt 2, \\dfrac{\\sqrt 2}2)$为椭圆$C_1$的一点, 与$OQ$平行的直线$l$与椭圆$C_1$交于两点$AB$, 若$\\triangle QAB$面积为$1$, 求直线$l$的方程;\\\\\n(3) 已知椭圆$C_1$与双曲线$C_2:x^2-y^2=1$在第一象限的交点为$M(x_M,y_M)$, 椭圆 $C_1$和双曲线$C_2$上满足$|x|\\ge |x_M|$的所有点$(x,y)$组成曲线$C$. 若点$N$是曲线$C$上一动点, 求$\\overrightarrow{NF_1}\\cdot \\overrightarrow{NF_2}$的取值范围.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "椭圆" ], "genre": "解答题", "ans": "", @@ -116007,7 +116517,8 @@ "content": "双曲线$\\dfrac{x^2}4-\\dfrac{y^2}9=1$的两渐近线的夹角的大小为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "双曲线" ], "genre": "填空题", "ans": "", @@ -116105,7 +116616,8 @@ "content": "已知$l_1,l_2$是分别经过点$A(2,1)$和$B(0,2)$两点的两条平行直线, 当$l_1,l_2$之间的距离最大时, 直线$l_1$的方程是\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "填空题", "ans": "", @@ -116418,7 +116930,8 @@ "content": "已知圆$C$过定点$A(0,1)$, 圆心$C$在抛物线$x^2=2y$上, $M,N$为圆$C$与$x$轴的交点.\\\\\n(1) 当圆心$C$是抛物线的顶点时, 求抛物线的准线被该圆截得的弦长;\\\\\n(2) 当圆心$C$在抛物线上运动时, $|MN|$是否为一定值? 证明你的结论;\\\\\n(3) 当圆心$C$在抛物线上运动时, 记$|AM|=m$, $|AN|=n$, 求$\\dfrac mn+\\dfrac nm$的最大值, 并求出此时圆$C$的方程.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "抛物线" ], "genre": "解答题", "ans": "", @@ -116727,7 +117240,8 @@ "content": "已知$P$为椭圆$\\dfrac{x^2}{4}+\\dfrac{y^2}2=1$上任意一点, $Q$与$P$关于$x$轴对称, $F_1$, $F_2$为椭圆的左、右焦点, 若有$\\overrightarrow{F_1P}\\cdot \\overrightarrow{F_2P}\\le 1$, 则向量$\\overrightarrow{F_1P}$与$\\overrightarrow{F_2Q}$的夹角的取值范围为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "椭圆" ], "genre": "填空题", "ans": "", @@ -116946,7 +117460,8 @@ "content": "已知抛物线$y^2=4x$, $F$为焦点, $P$为准线$l$上一动点, 线段$PF$与抛物线交于点$Q$, 定义$d(P)=\\dfrac{|FP|}{|FQ|}$.\\\\\n(1) 若点$P$坐标为$(-1,-\\dfrac 83)$, 求$d(P)$;\\\\\n(2) 求证: 存在常数$a$, 使得$2d(P)=|FP|+a$恒成立;\\\\\n(3) 设$P_1,P_2,P_3$为准线$l$上的三点, 且$|P_1P_2|=|P_2P_3|$, 试比较$d(P_1)+d(P_3)$与$2d(P_2)$的大小.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "抛物线" ], "genre": "解答题", "ans": "", @@ -118345,7 +118860,8 @@ "content": "设$A,B$是一条斜率为$4$的直线与抛物线$y^2=x$的两个交点, 则线段$AB$的中点的坐标可能是\\blank{50}(写出一个可能的点的坐标).", "objs": [], "tags": [ - "第七单元" + "第七单元", + "抛物线" ], "genre": "填空题", "ans": "", @@ -118393,7 +118909,8 @@ "content": "过点$P(2,3)$的直线$l$分别交$x$轴、$y$轴的正半轴于$A$、$B$两点, 则当$|PA|\\cdot|PB|$取到最小值时, $l$的方程为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "填空题", "ans": "", @@ -118522,7 +119039,8 @@ "content": "设$m$是正实数, 若椭圆$mx^2+(m+1)y^2=1$的两焦点的距离为$3$, 则$m$的值为\\bracket{20}.\n\\fourch{$\\dfrac{\\sqrt{13}-3}6$}{$\\dfrac{\\sqrt{21}-3}6$}{$\\dfrac 13$}{$\\dfrac{\\sqrt{33}-3}6$}", "objs": [], "tags": [ - "第七单元" + "第七单元", + "椭圆" ], "genre": "选择题", "ans": "", @@ -118683,7 +119201,8 @@ "content": "已知抛物线$C$的方程为$y^2=x$, 圆$M$的方程为$(x-2)^2+y^2=1$.\\\\\n(1) 设$P$是抛物线$C$上的动点, 证明: $P$在圆$M$外;\\\\\n(2) 设斜率为$1$的直线$l$与圆$M$相切, 且与抛物线$C$交于$Q_1,Q_2$两点, 求$|Q_1Q_2|$的值;\\\\\n(3) 设$A_1,A_2,A_3$是抛物线$C$上的三点, 直线$A_1A_2$, 直线$A_1A_3$均与圆$M$相切, 判断直线$A_2A_3$与圆$M$的位置关系, 说明理由.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "抛物线" ], "genre": "解答题", "ans": "", @@ -119085,7 +119604,8 @@ "content": "已知点$O$是坐标原点, 点$A(0,2)$点$P$是抛物线$y=4x^2$上的点, 则使得$OPA$是等腰三角形的点$P$为\\bracket{20}.\n\\fourch{$2$}{$4$}{$6$}{$8$}", "objs": [], "tags": [ - "第七单元" + "第七单元", + "抛物线" ], "genre": "选择题", "ans": "", @@ -119215,7 +119735,8 @@ "content": "已知双曲线$\\Gamma:\\dfrac{x^2}2-\\dfrac{y^2}4=1$的右顶点为$A$, 点$B$的坐标为$(1,\\sqrt 2)$.\\\\\n(1) 设双曲线$\\Gamma$的两条渐近线的夹角为$\\theta$, 求$\\cos\\theta$;\\\\\n(2) 设点$D$是双曲线$\\Gamma$上的动点, 若点$N$满足$\\overrightarrow{BN}=\\overrightarrow{ND}$, 求点$N$的轨迹方程;\\\\\n(3) 过点$B$的动直线$l$交双曲线$\\Gamma$于$PQ$两个不同的点, $M$为线段$PQ$的中点, 求直线$AM$的斜率的取值范围.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "双曲线" ], "genre": "解答题", "ans": "", @@ -119885,7 +120406,8 @@ "content": "若双曲线$x^2-\\dfrac{y^2}m=1$的渐近线方程为$y=\\pm 2x$, 则实数$m=$\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "双曲线" ], "genre": "填空题", "ans": "", @@ -120287,7 +120809,8 @@ "content": "如图, 椭圆$C:\\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1$($a>b>0$)的左、右焦点分别为$F_1$、$F_2$, 过右焦点$F_2$与$x$轴垂直的直线交椭圆于$MN$两点, 动点$P$、$Q$分别在直线$MN$与椭圆$C$上.已知$|F_1F_2|=2$, $\\triangle MNF_1$的周长为$4\\sqrt 2$.\\\\\n\\begin{center}\n \\begin{tikzpicture}[>=latex,scale = 1.5]\n \\draw [->] (-2,0) -- (2,0) node [below] {$x$};\n \\draw [->] (0,-1.3) -- (0,1.3) node [left] {$y$};\n \\draw (0,0) node [below left] {$O$};\n \\draw (0,0) ellipse ({sqrt(2)} and 1);\n \\draw (1,{-sqrt(2)/2}) node [below] {$N$}-- (1,{sqrt(2)/2}) node [above] {$M$};\n \\filldraw (-1,0) circle (0.03) node [below] {$F_1$} (1,0) circle (0.03) node [below right] {$F_2$};\n \\draw (-1,0) -- (-1,{sqrt(2)/2}) node [above] {$Q$} -- (1,-0.4) node [right] {$P$} -- cycle;\n \\end{tikzpicture}\n\\end{center}\n(1)\t求椭圆$C$的方程;\\\\\n(2)\t若线段$PQ$的中点在$y$轴上, 求三角形$F_1QP$的面积;\\\\\n(3)\t是否存在以$F_1Q$、$F_1P$为邻边的矩形$F_1PEQ$, 使得点$E$在椭圆$C$上? 若存在, 求出所有满足条件的点$Q$的横坐标; 若不存在, 说明理由.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "椭圆" ], "genre": "解答题", "ans": "", @@ -120402,7 +120925,8 @@ "content": "若直线$l_1:2x+my+1=0$与$l_2:y=3x-1$垂直, 则实数$m=$\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "填空题", "ans": "", @@ -120597,7 +121121,8 @@ "content": "已知抛物线$y^2=4x$, 斜率为$k$的直线$l$经过抛物线的焦点$F$, 与抛物线交于$P$、$Q$两点, 点$Q$关于$x$轴的对称点为$Q'$, 点$P$关于直线$x=1$的对称点为$P'$, 且满足$P'Q'\\perp PQ$, 则直线$l$的方程为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "抛物线" ], "genre": "填空题", "ans": "", @@ -120691,7 +121216,8 @@ "content": "椭圆$C:\\dfrac{x^2}{4}+\\dfrac{y^2}{3}=1$的左、右顶点分别为$A_1,A_2$, 点$P$在$C$上($P$不与$A_1,A_2$重合)且直线$PA_2$的斜率的取值范围是$[-2,-1]$, 那么直线$PA_1$斜率的取值范围是\\bracket{20}.\n\\fourch{$[\\dfrac 12,\\dfrac 34]$}{$[\\dfrac 38,\\dfrac 34]$}{$[\\dfrac 12, 1]$}{$[\\dfrac 34,1]$}", "objs": [], "tags": [ - "第七单元" + "第七单元", + "椭圆" ], "genre": "选择题", "ans": "", @@ -120815,7 +121341,8 @@ "content": "已知椭圆$C:\\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1$($a>b>0$), 过定点$T(t,0)$的直线交椭圆于$P,Q$两点, 其中$t\\in (0,a)$.\n\\begin{center}\n \\begin{tikzpicture}[>=latex]\n \\draw [->] (-3,0) -- (3,0) node [below] {$x$};\n \\draw [->] (0,-2) -- (0,2) node [left] {$y$};\n \\draw (0,0) node [below left] {$O$};\n \\draw [name path = ell] (0,0) ellipse (2 and {sqrt(3)});\n \\draw (1,0) node [above left] {$T$} coordinate (T);\n \\path (0.4,-1.8) coordinate (U) ($(U)!2!(T)$) coordinate (V);\n \\path [name path = line] (U) -- (V);\n \\path [name intersections = {of = ell and line, by = {P,Q}}];\n \\draw ($(P)!{-0.2}!(Q)$) -- (P) node [above right] {$P$} -- (Q) node [below right ] {$Q$} -- ($(Q)!{-0.1}!(P)$) ;\n \\draw (2.5,0) node [below] {$S$} coordinate (S);\n \\draw (P) -- (S) -- (Q);\n \\end{tikzpicture}\n\\end{center}\n(1) 若椭圆短轴长为$2\\sqrt{3}$且经过点$(-1,\\dfrac 32)$, 求椭圆方程;\\\\\n(2) 对(1)中的椭圆, 若$t=\\sqrt{3}$, 求$\\triangle OPQ$面积的最大值;\\\\\n(3) 在$x$轴上是否存在点$S(s,0)$使得$\\angle PST=\\angle QST$恒成立? 如果存在, 求出$s,t$的关系; 如果不存在, 说明理由.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "椭圆" ], "genre": "解答题", "ans": "", @@ -120913,7 +121440,8 @@ "content": "直线$l$的参数方程为$\\begin{cases} x=2+t, \\\\ y=1+2t, \\end{cases}$($t\\in \\mathbf{R}$), 则直线$l$的斜率为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "填空题", "ans": "", @@ -121336,7 +121864,8 @@ "content": "如图, 中心在原点$O$的椭圆$\\Gamma$ 的右焦点为$F(2\\sqrt 3,0)$, 长轴长为$8$. 椭圆$\\Gamma$上有两点$P,Q$, 连结$OP,OQ$, 记它们的斜率为$k_{OP}$、$k_{OQ}$, 且满足$k_{OP}\\cdot k_{OQ}=-\\dfrac 14$.\n\\begin{center}\n \\begin{tikzpicture}[>=latex,scale = 0.5]\n \\draw [->] (-5,0) -- (5,0) node [below] {$x$};\n \\draw [->] (0,-4) -- (0,4) node [left] {$y$};\n \\draw (0,0) node [below left] {$O$};\n \\draw [name path = directrix] ({4*sqrt(3)},-4) -- ({4*sqrt(3)},5.2);\n \\draw (0,0) ellipse (4 and 2);\n \\draw ({4*cos(acos(sqrt(7)-sqrt(3)))},{2*sin(acos(sqrt(7)-sqrt(3)))}) node [above] {$P$} coordinate (P);\n \\draw ({-4*sin(acos(sqrt(7)-sqrt(3)))},{2*cos(acos(sqrt(7)-sqrt(3)))}) node [above] {$R$} coordinate (R);\n \\draw ({4*sin(acos(sqrt(7)-sqrt(3)))},{-2*cos(acos(sqrt(7)-sqrt(3)))}) node [below] {$Q$} coordinate (Q);\n \\draw (0,0) -- (P) -- (R) -- (Q) -- (P);\n \\draw [name path = line1] (Q) -- ($(Q)!2.7!(P)$);\n \\draw [name path = line2] (R) -- ($(R)!1.7!(P)$);\n \\draw [name intersections = {of = line1 and directrix, by = N}] (N) node [right] {$N$};\n \\draw [name intersections = {of = line2 and directrix, by = M}] (M) node [right] {$M$};\n \\end{tikzpicture}\n\\end{center}\n(1)求椭圆$\\Gamma$的标准方程;\n(2)求证: ${{| OP |}^2}+{{| OQ |}^2}$为一定值, 并求出这个定值;\n(3)设直线$OQ$与椭圆$\\Gamma$的另一个交点为$R$, 直线$RP$ 和$PQ$分别与直线$x=4\\sqrt 3$ 交于点$M,N$, 若$\\triangle PQR$和$\\triangle PMN$的面积相等, 求点$P$的横坐标.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "椭圆" ], "genre": "解答题", "ans": "", @@ -121405,7 +121934,8 @@ "content": "若双曲线方程为${x^2}-\\dfrac{y^2}{16}=1$, 则该双曲线的渐近线方程为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "双曲线" ], "genre": "填空题", "ans": "", @@ -121874,7 +122404,8 @@ "content": "已知抛物线$y^2=4x$的焦点为$F$, 直线$l$交抛物线于不同的$A$、$B$两点.\\\\\n(1) 若直线$l$的方程为$y=x-1$, 求线段$AB$的长;\\\\\n(2) 若直线$l$经过点$P(-1,0)$, 点$A$关于$x$轴的对称点为$A'$, 求证: $A'$、$F$、$B$三点共线;\\\\\n(3) 若直线$l$经过点$M(8,-4)$, 抛物线上是否存在定点$N$, 使得以线段$AB$为直径的圆恒过点$N$? 若存在, 求出点$N$的坐标, 若不存在, 请说明理由.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "抛物线" ], "genre": "解答题", "ans": "", @@ -213338,7 +213869,8 @@ "content": "写出下列直线的一个方向向量$\\overrightarrow d$.\\\\\n(1) $2x-3y+1=0$;\\\\ \n(2) $3x+1=0$;\\\\\n(3) $4-2y=0$.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -213359,7 +213891,8 @@ "content": "求过点$P$且与$\\overrightarrow d$平行的直线$l$的点方向式方程.\\\\\n(1) $P(0,0)$, $\\overrightarrow d=(1,1)$;\\\\\n(2) $P(-2,3)$, $\\overrightarrow d=(-2,3)$.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -213380,7 +213913,8 @@ "content": "求经过$AB$两点的直线$l$的点方向式方程.\\\\\n(1) $A(3,5)$, $B(1,0)$;\\\\ \n(2) $A(2,2)$, $B(5,5)$.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -213401,7 +213935,8 @@ "content": "已知平行四边形$ABCD$的三个顶点的坐标分别为$A(1,2)$、$B(3,4)$、$C(2,6)$, 求四条边$AB,BC,CD$和$DA$所在直线的点方向式方程.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -213424,7 +213959,8 @@ "content": "求下列直线的一个法向量$\\overrightarrow n$.\\\\\n(1) $2x-3y+1=0$;\\\\\n(2) $\\dfrac x2+\\dfrac y5=1$.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -213445,7 +213981,8 @@ "content": "求经过点$P$且垂直于向量$\\overrightarrow n$的直线的点法向式方程.\\\\\n(1) $P(0,0)$, $\\overrightarrow n=(1,1)$;\\\\\n(2) $P(5,2)$, $\\overrightarrow n=(0,2)$.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -213466,7 +214003,8 @@ "content": "已知直线$\\dfrac{x-2}3=\\dfrac{y+1}2$的一个法向量为$\\overrightarrow n=(a,a-2)$, 求实数$a$的值.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -213487,7 +214025,8 @@ "content": "已知$\\triangle ABC$的三个顶点的坐标分别为$A(3,8)$、$B(3,-2)$、$C(-3,0)$.\\\\\n(1) 求$BC$边所在直线的方程;\\\\\n(2) 求$AB$边上中线$CM$所在直线的方程;\\\\\n(3) 求$BC$边上高$AD$所在直线的方程.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -213510,7 +214049,8 @@ "content": "已知四边形$ABCD$是平行四边形, $AB$边所在直线的方程是$x+y-1=0$, $AD$边所在直线的方程是$3x-y+4=0$, 顶点$C$的坐标是$(3,3)$, 求这个平行四边形其他两条边所在直线的方程.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -213531,7 +214071,8 @@ "content": "已知$\\triangle ABC$的两个顶点的坐标分别是$A(-2,1)$、$B(4,-3)$, 且$\\triangle ABC$的垂心坐标为$H(0,2)$, 分别求$BC,AC$边所在直线的方程.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -213554,7 +214095,8 @@ "content": "已知$\\triangle ABC$的三个顶点的坐标分别为$A(2,1)$、$B(0,7)$、$C(-4,-1)$.\\\\\n(1) 求此三角形的三边所在直线的方程;\\\\\n(2) 求此三角形的三条中线所在直线的方程.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -213575,7 +214117,8 @@ "content": "已知$\\triangle ABC$的三个顶点的坐标分别$A(4,0)$、$B(6,7)$、$C(0,3)$, 求此三角形的三条高所在直线的方程.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -213596,7 +214139,8 @@ "content": "已知梯形$ABCD$的三个顶点的坐标分别为$A(2,3)$、$B(-2,1)$、$C(4,5)$, 求此梯形中位线所在直线的方程.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -213619,7 +214163,8 @@ "content": "已知$A(1,-1)$、$B(3,3)$两点, 点$C(5,a)$在直线$AB$上, 求实数$a$的值.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -213640,7 +214185,8 @@ "content": "填空:\n方程$\\begin{vmatrix} x+1 & y-1 \\\\4 & 1 \\end{vmatrix}=0$所表示的直线的一个法向量是\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "填空题", "ans": "", @@ -213661,7 +214207,8 @@ "content": "已知原点$O$在直线$l$上的射影为$H(-2,1)$, 求直线$l$的方程.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -213705,7 +214252,8 @@ "content": "已知$\\triangle ABC$的两个顶点的坐标分别是$A(2,2)$、$B(3,0)$, 此三角形的重心坐标为$(3,1)$.\\\\\n(1) 求此三角形的三边所在直线的方程;\\\\\n(2) 求此三角形的三条中线所在直线的方程.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -213726,7 +214274,8 @@ "content": "已知$a,b,c$是互不相等的正实数, 求经过下列两点的直线的倾斜角:\\\\\n(1) $A(a,c),B(b,c)$;\\\\\n(2) $C(a,\\sqrt {a^2+b^2}),D(a,\\sqrt {b^2+c^2})$;\\\\\n(3) $E(b,b+c),F(a,a+c)$;\\\\\n(4) $G(b,a^2),H(a,b^2)$.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -213747,7 +214296,8 @@ "content": "已知斜率为$3$的直线过点$(1,1)$和$(x,-2)$, 求实数$x$的值.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -213768,7 +214318,8 @@ "content": "求下列方程所表示的直线的斜率:\\\\\n(1) $x+5=0$;\\\\\n(2) $2y+3=0$;\\\\\n(3) $\\dfrac{x-3}3=\\dfrac{y+1}{-4}$;\\\\\n(4) $5x+6y+3=0$.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -213789,7 +214340,8 @@ "content": "已知直线$l$的倾斜角为$\\alpha$, 且这条直线经过点$P(3,5)$, 求直线$l$的方程.\\\\\n(1) $\\alpha =\\dfrac{\\pi }4$;\\\\ \n(2) $\\alpha =0$;\\\\\n(3) $\\alpha =\\dfrac{\\pi}2$.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -213810,7 +214362,8 @@ "content": "写出经过下列两点的直线的点斜式方程.\\\\\n(1) $A(-3,0),B(2,-2)$;\\\\ \n(2) $P(2,-2),Q(0,1)$.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -213831,7 +214384,8 @@ "content": "求过点$(\\sqrt 3,\\sqrt 5)$, 倾斜角等于直线$y=\\sqrt 3x+1$的倾斜角的一半的直线方程.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -213852,7 +214406,8 @@ "content": "已知直线$l$的倾斜角为$\\alpha$, $\\sin \\alpha =\\dfrac 35$, 且这条直线经过点$P(3,5)$, 求直线$l$的一般式方程.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -213873,7 +214428,8 @@ "content": "已知矩形$OACB$的顶点的坐标分别为$O(0,0)$、$A(8,0)$、$B(0,5)$, 求该矩形的对角线所在直线的方程.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -213894,7 +214450,8 @@ "content": "已知直线$2x-3y+6=0$, 这条直线的点方向式方程可以是\\bracket{20}.\n\\fourch{$\\dfrac{x-3}2=\\dfrac{y-4}3$}{$\\dfrac x{-2}=\\dfrac{y-2}3$}{$\\dfrac{x+3}3=\\dfrac y2$}{$\\dfrac{x+3}2=\\dfrac y3$}", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "选择题", "ans": "", @@ -213915,7 +214472,8 @@ "content": "求过点$P$且平行于直线$l_0$的直线的一般式方程.\\\\\n(1) $P(2,1)$, $l_0:x+4=0$;\\\\ \n(2) $P(1,2)$, $l_0:\\dfrac x3+\\dfrac y4+\\dfrac 17=0$.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -213936,7 +214494,8 @@ "content": "求过点$P$且垂直于直线$l_0$的直线的一般式方程.\\\\\n(1) $P(2,1)$, $l_0:y-3=0$;\\\\ \n(2) $P(-2,-1)$, $l_0:\\dfrac{x-1}3=\\dfrac{y+2}4$.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -213957,7 +214516,8 @@ "content": "分别根据下列条件, 求相应直线$l$的方程.\\\\\n(1) 直线$l$经过$A(2,0)$、$B(3,7)$两点;\\\\\n(2) 直线$l$经过点$P(3,4)$, 且与向量$\\overrightarrow d=(1,-1)$平行;\\\\\n(3) 直线$l$与$x$轴交于点$A(3,0)$, 与$y$轴交于点$B(0,-1)$;\\\\\n(4) 直线$l$经过点$P(3,2)$, 且与向量$\\overrightarrow n=(8,-4)$垂直.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -213978,7 +214538,8 @@ "content": "选择题:\n直线$x-ay+2=0(a<0)$的倾斜角是\\bracket{20}.\n\\fourch{$\\arctan \\dfrac 1a$}{$-\\arctan \\dfrac 1a$}{$\\pi -\\arctan \\dfrac 1a$}{$\\pi +\\arctan \\dfrac 1a$}", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "选择题", "ans": "", @@ -213999,7 +214560,8 @@ "content": "当$\\theta \\in [-\\dfrac{\\pi }2,0)$时, 求经过$P(0,0)$、$Q(\\cos \\theta ,\\sin \\theta)$两点的直线的斜率.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -214020,7 +214582,8 @@ "content": "已知$\\triangle ABC$的顶点坐标分别为$A(-3,0)$、$B(1,2)$、$C(3,9)$, 直线$l$过顶点$C$, 且把$\\triangle ABC$分为面积相等的两部分, 求直线$l$的方程.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -214041,7 +214604,8 @@ "content": "已知直线$l$经过点$A(3,4)$, 它的倾斜角是直线$2x-y+1=0$的倾斜角的$2$倍, 求直线$l$的方程.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -214083,7 +214647,8 @@ "content": "一根金属棒在$30^\\circ\\text{C}$时长$14.205$米, 在$60^\\circ\\text{C}$时长$14.211$米.已知长度$y$(米)和温度$x$($^\\circ\\text{C}$)的关系可以用直线方程来表示, 试写出这条直线的方程, 并求$80^\\circ\\text{C}$时这根金属棒的长度.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -214104,7 +214669,8 @@ "content": "判断下列各组直线的位置关系:\\\\\n(1) $l_1$: $2x-3y-1=0$, $l_2$: $4x-6y-2=0$;\\\\\n(2) $l_1$: $\\dfrac x3-\\dfrac{12}5y+1=0$, $l_2$: $36x+5y-1=0$;\\\\\n(3) $l_1$: $y=\\dfrac 13(x-6)$, $l_2$: $3x+y-3=0$;\\\\\n(4) $l_1$: $(\\sqrt 5+1)x-2y+1=0$, $l_2$: $2x-(\\sqrt 5-1)y-1=0$.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -214125,7 +214691,8 @@ "content": "已知直线$l_1$: $6x+(t-1)y-8=0$与直线$l_2$: $(t+4)x+(t+6)y-16=0$.\\\\\n(1) 当$t$为何值时, $l_1$与$l_2$相交?\\\\\n(2) 当$t$为何值时, $l_1$与$l_2$平行?\\\\\n(3) 当$t$为何值时, $l_1$与$l_2$重合?\\\\\n(4) 当$t$为何值时, $l_1$与$l_2$垂直?", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -214146,7 +214713,8 @@ "content": "已知直线$l_1$: $mx+8y+n=0$与直线$l_2$: $2x+my-1=0$.当直线$l_1$与直线$l_2$分别满足下列条件时, 求实数$mn$的值.\\\\\n(1) 直线$l_1$与直线$l_2$平行;\\\\\n(2) 直线$l_1$与直线$l_2$垂直, 且直线$l_1$在$y$轴上的截距为$-1$.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -214167,7 +214735,8 @@ "content": "根据下列条件, 写出满足条件的直线的一般式方程.\\\\\n(1) 经过直线$2x-y+1=0$与直线$2x+2y-1=0$的交点, 且与直线$5x-y=0$垂直;\\\\\n(2) 经过直线$x-y+1=0$与直线$2x-y+2=0$的交点, 且与直线$3x+4y=12$平行.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -214188,7 +214757,8 @@ "content": "已知直线$l_1$: $y=kx+k+2$与直线$l_2$: $y=-2x+4$的交点在第一象限, 求实数$k$的范围.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -214230,7 +214800,8 @@ "content": "是否存在实数$a$, 使直线$l_1$: $(a-1)x+(a-2)y-1=0$与直线$l_2$: $6x+(2a-3)y-3=0$平行? 若存在, 求$a$的值; 若不存在, 请说明理由.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -214251,7 +214822,8 @@ "content": "若直线$mx-2y=1$与直线$6x-4y+n=0$重合, 求实数$m,n$的值.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -214274,7 +214846,8 @@ "content": "若直线$(2a^2-7a+3)x+(a^2-9)y+3=0$与$x$轴平行, 求实数$a$的值.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -214295,7 +214868,8 @@ "content": "求下列两组直线的夹角:\\\\\n(1) $l_1$: $\\sqrt 3x-y=0$, $l_2$: $x-\\sqrt 3y+2=0$;\\\\\n(2) $l_1$: $x-1=0$, $l_2$: $x+y-5=0$.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -214316,7 +214890,8 @@ "content": "已知直线$\\sqrt 3x+y=0$与直线$kx-y+1=0$的夹角为$60^{\\circ }$, 求实数$k$的值.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -214337,7 +214912,8 @@ "content": "已知等腰直角三角形$ABC$的斜边$AB$所在直线的方程为$3x-y-5=0$, 直角顶点为$C(4,-1)$, 求两条直角边所在直线的方程.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -214360,7 +214936,8 @@ "content": "判断直线$l_1$与直线$l_2$的位置关系.\\\\\n(1) $l_1$: $2x-3=0$, $l_2$: $ax+y-1=0$, 其中$a\\in \\mathbf{R}$;\\\\\n(2) $l_1$: $(a^2+1)x+y-1=0$, $l_2$: $(2a^2+3)x+y+a=0$, 其中$a\\in \\mathbf{R}$.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -214381,7 +214958,8 @@ "content": "已知直线$y=ax+b$的倾斜角为$\\dfrac{3\\pi }4$, 且这条直线与直线$5x+3y-31=0$的交点在第一象限内, 求实数$b$的取值范围.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -214402,7 +214980,8 @@ "content": "已知直线$l_1$: $ax+by+4=0$, 直线$l_2$: $(1-a)x-y-b=0$, 直线$l_3$: $x+2y+3=0$, 且这三条直线两两平行, 求实数$ab$的值.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -214423,7 +215002,8 @@ "content": "已知等腰直角三角形$ABC$的直角边$BC$所在直线的方程为: $x-2y-6=0$, 顶点$A$的坐标为$(0,6)$, 求斜边$AB$和直角边$AC$所在直线的方程.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -214446,7 +215026,8 @@ "content": "已知直线$l_1$: $a_1x+b_1y+c_1=0$(实数$a_1,b_1$不同时为零), 直线$l_2$: $a_2x+b_2y+c_2=0$(实数$a_2,b_2$不同时为零), 用$l_1$与$l_2$的法向量求这两条直线的夹角.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -214467,7 +215048,8 @@ "content": "求点$P(3,2)$到直线$l$: $3x-2y=13$的距离.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -214488,7 +215070,8 @@ "content": "已知$P(0,0)$、$Q(3,2)$两点, 试判断$P,Q$是否在下列直线的同一侧.\\\\\n(1) $2x+3y=4$;\\\\ \n(2) $x+3y+4=0$;\\\\\n(3) $2x-3y=4$;\\\\ \n(4) $-2x-3y+3=0$.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -214509,7 +215092,8 @@ "content": "已知直线$l_1$: $2x-y+a=0$与直线$l_2$: $-4x+2y+1=0$, 且直线$l_1$与直线$l_2$的距离为$\\dfrac{7\\sqrt 5}{10}$, 求实数$a$的值.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -214532,7 +215116,8 @@ "content": "已知点$P$为直线$3x-4y+2=0$上的任意一个动点, 求点$P$到点$A(3,-1)$的距离的最小值.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -214553,7 +215138,8 @@ "content": "已知$A(2,3)$、$B(-4,8)$两点, 直线$l$经过原点, 且$AB$两点到直线$l$的距离相等, 求直线$l$的方程.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -214574,7 +215160,8 @@ "content": "已知平行直线$l_1$与$l_2$的距离为$\\sqrt 5$, 且直线$l_1$经过原点, 直线$l_2$经过点$(1,3)$, 求直线$l_1$和直线$l_2$的方程.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -214595,7 +215182,8 @@ "content": "已知直线$l$过点$P(0,1)$, 且被平行直线$l_1$: $3x+4y-8=0$与$l_2$: $3x+4y+2=0$所截得的线段的长为$2\\sqrt 2$, 求直线$l$的方程.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -214616,7 +215204,8 @@ "content": "求与直线$x-2y+1=0$和$2x-y+3=0$距离相等的点的轨迹.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -214637,7 +215226,8 @@ "content": "已知正方形$ABCD$的中心的坐标为点$P(1,1)$, $AB$边所在直线的方程为$x+2y+3=0$.求这个正方形的其他三边所在直线的方程.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -214658,7 +215248,8 @@ "content": "求原点$O$到直线$x\\cos \\theta +y\\sin \\theta +2=0$, $\\theta \\in \\mathbf{R}$的距离.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -214679,7 +215270,8 @@ "content": "已知直线$l$过点$(2,4)$, 且它被平行直线$l_1$: $x-y+1=0$与直线$l_2$: $x-y-2=0$所截得的线段的中点在直线$l_3$: $x+2y-3=0$上, 求直线$l$的方程.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -214700,7 +215292,8 @@ "content": "已知$P_1(1,0)$、$P_{2}(7,-8)$两点分别在直线$l$的两侧, 且$P_1,P_2$到直线$l$的距离均为$4$, 求直线$l$的方程.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -214723,7 +215316,8 @@ "content": "若直线$l$过点$P(0,2)$, 它的一个方向向量为$(1,1)$, 则直线$l$的方程是\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "填空题", "ans": "", @@ -214744,7 +215338,8 @@ "content": "若直线$l$过点$(3,1)$, 且$l$的法向量$\\overrightarrow n=(1,3)$, 则直线$l$的方程为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "填空题", "ans": "", @@ -214765,7 +215360,8 @@ "content": "经过点$P(x_0,y_0)$, 且与向量$\\overrightarrow d=(u,v)$平行的直线方程是\\bracket{20}.\n\\fourch{$\\dfrac{x-x_0}u=\\dfrac{y-y_0}v$}{$\\dfrac{x-{x_0}}{y-{y_0}}=\\dfrac uv$}{$y-y_0=\\dfrac vu(x-x_0)$}{$u(y-y_0)=v(x-x_0)$}", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "选择题", "ans": "", @@ -214786,7 +215382,8 @@ "content": "若直线$x=1$的倾斜角为$\\theta$, 则$\\theta$等于\\bracket{20}.\n\\fourch{$0$}{$\\dfrac{\\pi }4$}{$\\dfrac{\\pi }2$}{不存在}", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "选择题", "ans": "", @@ -214807,7 +215404,8 @@ "content": "如图, 已知矩形$OABC$的顶点$A$的坐标为$(5,2)$, 求直线$AB$的方程.\n\\begin{center}\n \\begin{tikzpicture}[>=latex, scale = 0.5]\n \\draw [->] (-2,0) -- (6,0) node [below] {$x$};\n \\draw [->] (0,-1) -- (0,4) node [left] {$y$};\n \\draw (0,0) node [below left] {$O$};\n \\draw (0,0) -- (5,2) node [right] {$A$} --++ (-1,2.5) node [above] {$B$} --++ (-5,-2) node [left] {$C$} -- cycle;\n \\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -214828,7 +215426,8 @@ "content": "已知直线$x-ay-4=0$与直线$y=-2x+4$的夹角$\\theta =\\arccos \\dfrac{2\\sqrt 5}5$, 求实数$a$的值.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -214851,7 +215450,8 @@ "content": "已知直线$l$经过点$(5,10)$, 且它与原点的距离为$5$, 求直线$l$的方程.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -214872,7 +215472,8 @@ "content": "已知直线$x-ay=0(a\\ge 0)$, 求这条直线的倾斜角.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -214893,7 +215494,8 @@ "content": "是否存在实数$m$, 使直线$l_1$: $(m+3)x+5y=5-3m$与直线$l_2$: $2x+(m+6)y=8$分别相交、平行、重合、垂直? 若存在, 求$m$的值; 若不存在, 请说明理由.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -214914,7 +215516,8 @@ "content": "已知$\\triangle ABC$的$AB,AC$边上的高所在直线的方程分别为$2x-3y+1=0$和$x+y=0$, 点$A$的坐标为$(1,2)$, 求$BC$边所在直线的方程.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -214935,7 +215538,8 @@ "content": "已知直线$l$垂直于直线$3x+4y-9=0$, 且点$A(2,3)$到直线$l$的距离为$1$, 求直线$l$的方程.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -214958,7 +215562,8 @@ "content": "已知直线$l_1$: $x+a^2y+1=0$的方向向量与直线$l_2$: $(a^2+1)x-by+3=0$的法向量平行, 且$a\\cdot b\\ne 0$, 求$|ab|$的最小值.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -214979,7 +215584,8 @@ "content": "求证: 三条互不平行的直线$l_1$: $a_1x+b_1y+c_1=0$, 直线$l_2$: $a_2x+b_2y+c_2=0$, 直线$l_3$: $a_3x+b_3y+c_{3}=0$共点的充要条件是$\\begin{vmatrix}\n a_1 & b_1 & c_1 \\\\a_2 & b_2 & c_2 \\\\a_3 & b_3 & c_3 \\end{vmatrix}=0$.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -215000,7 +215606,8 @@ "content": "求直线$l_1$: $3x-2y-6=0$关于直线$l$: $2x-3y+1=0$对称的直线$l_2$的方程.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -215023,7 +215630,8 @@ "content": "已知两条平行直线分别过点$P(-2,-2)$、$Q(1,3)$, 当这两条直线之间的距离最大时, 求它们的方程.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -215044,7 +215652,8 @@ "content": "如图, $\\angle BAC$为伸入江中的半岛, $AB$和$AC$为两江岸, $M$处为水文站, $N$处为电讯局, 现欲在两江岸$AB$和$AC$上各建一个水文观测点$PQ$.现测得$\\angle BAC=45^{\\circ }$, 当直角坐标系以点$A$为坐标原点且以直线$BA$为$x$轴时, 测得$M(-4,1)N(-3,2)$.$PQ$两点应建在何处才能使路程$MPQN$最短?\n\\begin{center}\n \\begin{tikzpicture}[>=latex]\n \\draw [->] (-3,0) -- (1,0) node [below] {$x$};\n \\draw [->] (0,-1) -- (0,3) node [left] {$y$};\n \\draw (0,0) node [below right] {$(O)$};\n \\draw (0,0) node [below left] {$A$};\n \\draw (-2,0.5) node [left] {$M$} (-1.5,1) node [left] {$N$};\n \\draw (-2.5,2.5) node [left] {$C$} -- (0,0);\n \\draw (-3,0) node [below] {$B$};\n \\draw (-1.5,1) -- (-1,1) node [above right] {$Q$} -- (-1.7,0) node [below] {$P$} -- (-2,0.5);\n \\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -215065,7 +215674,8 @@ "content": "已知直线$l$: $f(x,y)=0$.如果直线$l$外一点$P$的坐标为$(x_0,y_0)$, 那么直线$f(x,y)$ $-f(x_0,y_0)=0$\\bracket{20}.\n\\twoch{过点$P$且与直线$l$斜交}{过点$P$且与直线$l$重合}{过点$P$且与直线$l$平行}{过点$P$且与直线$l$垂直}", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "选择题", "ans": "", @@ -215086,7 +215696,8 @@ "content": "如果直线$x\\cos \\theta +y-2=0(\\theta \\in \\mathbf{R})$的倾斜角为$\\alpha$, 那么$\\alpha$的取值范围是\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "填空题", "ans": "", @@ -215107,7 +215718,8 @@ "content": "若直线$l_1$: $a_1x+b_1y+2=0$(实数$a_1,b_1$不同时为$0$)与直线$l_2$: $a_2x+b_2y+2=0$(实数$a_2,b_2$不同时为$0$)的交点为$(1,2)$, 则经过$P(a_1,b_1),Q(a_2,b_2)$两点的直线的方程为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "填空题", "ans": "", @@ -215128,7 +215740,8 @@ "content": "如果直线$l$经过点$(3,4)$, 且点$(-3,2)$到直线$l$的距离最大, 求这条直线的方程.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -215149,7 +215762,8 @@ "content": "已知$\\triangle ABC$的三个顶点的坐标分别为$A(2,3)$、$B(4,-1)$、$C(-4,1)$, 直线$l$平行于$AB$, 且将$\\triangle ABC$分成面积相等的两部分, 求直线$l$的方程.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -215170,7 +215784,8 @@ "content": "过点$P(2,1)$作直线$l$, 分别交$x$轴、$y$轴的正半轴于$AB$两点.当$\\triangle AOB$的面积最小时, 求直线$l$的方程.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -215191,7 +215806,8 @@ "content": "已知直线$l$经过点$P(1,2)$, 且与两坐标轴围成的三角形面积为$S$.\\\\\n(1) 当$S=3$时, 满足条件的直线有几条?\\\\\n(2) 当$S=4$时, 满足条件的直线有几条?\\\\\n(3) 当$S=5$时, 满足条件的直线有几条?", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -216038,7 +216654,8 @@ "content": "写出分别满足下列条件的椭圆的标准方程.\\\\\n(1) 焦点坐标为$(-6,0)$、$(6,0)$, 且椭圆经过点$(0,8)$;\\\\\n(2) 椭圆经过$(0,-2)$、$(\\sqrt 6,0)$两点;\\\\\n(3) 焦距等于$4$, 且椭圆经过点$P(\\dfrac{2\\sqrt 6}3,-\\dfrac{2\\sqrt 6}3)$.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "椭圆" ], "genre": "解答题", "ans": "", @@ -216080,7 +216697,8 @@ "content": "若方程$\\dfrac{x^2}m+\\dfrac{y^2}{{m^2}-2}=1$表示椭圆, 求实数$m$的取值范围.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "椭圆" ], "genre": "解答题", "ans": "", @@ -216101,7 +216719,8 @@ "content": "若椭圆$\\dfrac{x^2}{25}+\\dfrac{y^2}9=1$的两个焦点分别为$F_1F_2$, 点$P$为此椭圆上的任意一点, 求$\\triangle PF_1F_2$的周长.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "椭圆" ], "genre": "解答题", "ans": "", @@ -216122,7 +216741,8 @@ "content": "已知椭圆的方程为$\\dfrac{x^2}{16}+\\dfrac{y^2}{m^2}=1(m>0)$.如果此椭圆的焦点在$x$轴上, 那么它的焦距为\\bracket{20}.\n\\fourch{$2\\sqrt {16-m^2}$}{$2\\sqrt {4-m}$}{$2\\sqrt {m^2-8}$}{$2\\sqrt {m-4}$}", "objs": [], "tags": [ - "第七单元" + "第七单元", + "椭圆" ], "genre": "选择题", "ans": "", @@ -216143,7 +216763,8 @@ "content": "若方程$16x^2+ky^2=16k$表示焦点在$y$轴上的椭圆, 则实数$k$满足\\bracket{20}.\n\\fourch{$k>16$}{$k=16$}{$k<16$}{$0b>0)$上, 则\\bracket{20}.\n\\twoch{点$(4,-3)$不在椭圆上}{点$(3,4)$在椭圆上}{点$(-4,-3)$不在椭圆上}{点$(-4,3)$在椭圆上}", "objs": [], "tags": [ - "第七单元" + "第七单元", + "椭圆" ], "genre": "选择题", "ans": "", @@ -216271,7 +216897,8 @@ "content": "过点$(3,-2)$且与椭圆$4x^2+9y^2=36$有相同焦点的椭圆的标准方程是\\bracket{20}.\n\\fourch{$\\dfrac{x^2}{15}+\\dfrac{y^2}{10}=1$}{$\\dfrac{x^2}{{{15}^2}}+\\dfrac{y^2}{{{10}^2}}=1$}{$\\dfrac{x^2}{10}+\\dfrac{y^2}{15}=1$}{$\\dfrac{x^2}{{{10}^2}}+\\dfrac{y^2}{{{15}^2}}=1$}", "objs": [], "tags": [ - "第七单元" + "第七单元", + "椭圆" ], "genre": "选择题", "ans": "", @@ -216292,7 +216919,8 @@ "content": "若椭圆$\\dfrac{x^2}9+\\dfrac{y^2}4=1$的弦$AB$被点$P(1,1)$平分, 则$AB$所在直线的方程为 \\bracket{20}.\n\\fourch{$9x+4y-13=0$}{$4x+9y-13=0$}{$x+2y-3=0$}{$x+3y-3=0$}", "objs": [], "tags": [ - "第七单元" + "第七单元", + "椭圆" ], "genre": "选择题", "ans": "", @@ -216313,7 +216941,8 @@ "content": "画出长轴长和短轴长分别为$2$厘米、$1.5$厘米的椭圆的草图.若要把一个边长分别为$2$米和$1.5$米的矩形木板锯成椭圆形, 使它的长轴长和短轴长分别为$2$米、$1.5$米, 请用简便的方法在木板上画出这个椭圆的草图.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "椭圆" ], "genre": "解答题", "ans": "", @@ -216334,7 +216963,8 @@ "content": "已知椭圆以原点为中心, 长轴长是短轴长的$2$倍, 且过点$(-2,-4)$, 求此椭圆的标准方程.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "椭圆" ], "genre": "解答题", "ans": "", @@ -216376,7 +217006,8 @@ "content": "已知椭圆的一个顶点和一个焦点分别是直线$x+3y-6=0$与两坐标轴的交点, 求此椭圆的标准方程.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "椭圆" ], "genre": "解答题", "ans": "", @@ -216397,7 +217028,8 @@ "content": "已知椭圆$C$的焦点分别为$F_1(-2\\sqrt 2,0)$、$F_2(2\\sqrt 2,0)$, 长轴长为$6$, 直线$y=x+2$交椭圆$C$于$AB$两点, 求线段$AB$的中点的坐标.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "椭圆" ], "genre": "解答题", "ans": "", @@ -216418,7 +217050,8 @@ "content": "已知点$P$是椭圆$\\dfrac{x^2}{100}+\\dfrac{y^2}{36}=1$上一点, 它到椭圆的左焦点$F_1$的距离是它到右焦点$F_2$的距离的$3$倍.\\\\\n(1) 分别求点$P$与点$F_1$、点$P$与点$F_2$的距离;\\\\\n(2) 求点$P$的坐标.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "椭圆" ], "genre": "解答题", "ans": "", @@ -216439,7 +217072,8 @@ "content": "以椭圆$\\dfrac{x^2}{25}+\\dfrac{y^2}{16}=1$的两个焦点及短轴的两个端点为四个顶点的椭圆方程为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "椭圆" ], "genre": "填空题", "ans": "", @@ -216460,7 +217094,8 @@ "content": "如果点$P$是椭圆$\\dfrac{x^2}{36}+\\dfrac{y^2}{20}=1$上一个动点, $F_1$是椭圆的左焦点, 那么$|PF_1|$的最大值是\\blank{50}, $|PF_1|$的最小值是\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "椭圆" ], "genre": "填空题", "ans": "", @@ -216481,7 +217116,8 @@ "content": "如果直线$y=kx+1$与椭圆$\\dfrac{x^2}5+\\dfrac{y^2}m=1$恒有公共点, 那么实数$m$的取值范围为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "椭圆" ], "genre": "填空题", "ans": "", @@ -216502,7 +217138,8 @@ "content": "椭圆$\\dfrac{x^2}{25}+\\dfrac{y^2}9=1$与$\\dfrac{x^2}{9-k}+\\dfrac{y^2}{25-k}=1(0b>0)$与直线$x+2y-2=0$交于$AB$两点, $|AB|=\\sqrt 5$, 且$AB$的中点的坐标为$(m,\\dfrac 12)$, 求此椭圆的方程.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "椭圆" ], "genre": "解答题", "ans": "", @@ -216609,7 +217248,8 @@ "content": "写出分别满足下列条件的双曲线的标准方程.\\\\\n(1) 曲线上的点$P$到点$F_1(4,0)$的距离与它到点$F_2(-4,0)$的距离的差的绝对值等于$6$;\\\\\n(2) 曲线上的点$P$到点$F_1(-10,0)$的距离与它到点$F_2(10,0)$的距离的差等于$16$;\\\\\n(3) 焦点在$x$轴上, 且双曲线经过点$(-\\sqrt 2,-\\sqrt 3)$、$(\\dfrac{\\sqrt {15}}3,\\sqrt 2)$.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "双曲线" ], "genre": "解答题", "ans": "", @@ -216630,7 +217270,8 @@ "content": "设方程$\\dfrac{x^2}{m+2}-\\dfrac{y^2}{m+1}=1$表示焦点在$y$轴上的双曲线, 求实数$m$的取值范围.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "双曲线" ], "genre": "解答题", "ans": "", @@ -216655,7 +217296,8 @@ "content": "已知双曲线的对称轴为坐标轴, 焦点为$(-6,0)$、$(6,0)$, 且双曲线经过点$(-5,2)$, 求此双曲线的标准方程.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "双曲线" ], "genre": "解答题", "ans": "", @@ -216676,7 +217318,8 @@ "content": "过双曲线$\\dfrac{x^2}{16}-\\dfrac{y^2}9=1$的右焦点$F_2$作$x$轴的垂线, 求此垂线与双曲线的交点$M$到左焦点$F_1$的距离.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "双曲线" ], "genre": "解答题", "ans": "", @@ -216697,7 +217340,8 @@ "content": "已知双曲线关于原点对称, 它的焦点在坐标轴上, 焦距为$10$, 且此双曲线经过点$(3,4\\sqrt 2)$, 求它的标准方程.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "双曲线" ], "genre": "解答题", "ans": "", @@ -216718,7 +217362,8 @@ "content": "已知双曲线$\\dfrac{x^2}{64}-\\dfrac{y^2}{36}=1$的左、右焦点分别为$F_1,F_2$, 直线$l$过点$F_1$, 交双曲线的左支于$A,B$两点, 且$|AB|=m$, 求$\\triangle ABF_2$的周长.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "双曲线" ], "genre": "解答题", "ans": "", @@ -216739,7 +217384,8 @@ "content": "如果中心在原点, 对称轴在坐标轴上的等轴双曲线的一个焦点为$F_1(0,-6)$, 那么此双曲线的标准方程是\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "双曲线" ], "genre": "填空题", "ans": "", @@ -216760,7 +217406,8 @@ "content": "双曲线$2x^2-y^2=8$的焦点坐标是\\blank{50}, 两条渐近线的夹角为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "双曲线" ], "genre": "填空题", "ans": "", @@ -216781,7 +217428,8 @@ "content": "若双曲线的中心在坐标原点, 它的一个焦点的坐标是$(-5,0)$, 两个顶点间的距离为$6$, 则此双曲线的方程是\\bracket{20}.\n\\fourch{$\\dfrac{x^2}9-\\dfrac{y^2}{16}=1$}{$\\dfrac{x^2}{36}-\\dfrac{y^2}{11}=1$}{$\\dfrac{x^2}{16}-\\dfrac{y^2}9=1$}{$\\dfrac{x^2}{11}-\\dfrac{y^2}{36}=1$}", "objs": [], "tags": [ - "第七单元" + "第七单元", + "双曲线" ], "genre": "选择题", "ans": "", @@ -216802,7 +217450,8 @@ "content": "在下列双曲线中, 以$y=\\pm \\dfrac 12x$为渐近线的是\\bracket{20}.\n\\fourch{$\\dfrac{x^2}{16}-\\dfrac{y^2}4=1$}{$\\dfrac{x^2}4-\\dfrac{y^2}{16}=1$}{$\\dfrac{x^2}2-y^2=1$}{$x^2-\\dfrac{y^2}2=1$}", "objs": [], "tags": [ - "第七单元" + "第七单元", + "双曲线" ], "genre": "选择题", "ans": "", @@ -216825,7 +217474,8 @@ "content": "若方程$4x^2+ky^2=4k$表示双曲线, 则此双曲线的虚轴长等于\\bracket{20}.\n\\fourch{$2\\sqrt k$}{$2\\sqrt {-k}$}{$\\sqrt k$}{$\\sqrt {-k}$}", "objs": [], "tags": [ - "第七单元" + "第七单元", + "双曲线" ], "genre": "选择题", "ans": "", @@ -216867,7 +217517,8 @@ "content": "已知双曲线的虚轴的长为$6$, 一条渐近线的方程为$3x-y=0$, 求此双曲线的标准方程.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "双曲线" ], "genre": "解答题", "ans": "", @@ -216888,7 +217539,8 @@ "content": "求与双曲线$x^2-\\dfrac{y^2}4=1$有共同渐近线, 且过点$M(2,2)$的双曲线的标准方程.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "双曲线" ], "genre": "解答题", "ans": "", @@ -216909,7 +217561,8 @@ "content": "已知双曲线$\\dfrac{x^2}8-\\dfrac{y^2}{b^2}=1$的右焦点为点$F$, 若直线$x-y-3=0$经过点$F$, 求此双曲线渐近线的方程.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "双曲线" ], "genre": "解答题", "ans": "", @@ -216930,7 +217583,8 @@ "content": "已知双曲线$\\dfrac{x^2}9-\\dfrac{y^2}{16}=1$的两个焦点分别为$F_1,F_2$, 点$P$为此双曲线上一点, $|PF_1|\\cdot|PF_2|=32$, 求证: $PF_1\\perp PF_2$.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "双曲线" ], "genre": "解答题", "ans": "", @@ -216953,7 +217607,9 @@ "content": "求以椭圆$\\dfrac{x^2}8+\\dfrac{y^2}5=1$的焦点为顶点, 以椭圆的顶点为焦点的双曲线的方程.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "椭圆", + "双曲线" ], "genre": "解答题", "ans": "", @@ -217016,7 +217672,8 @@ "content": "已知直线$l$: $y=ax+1$与双曲线$C$: $3x^2-y^2=1$相交于$AB$两点.\\\\\n(1) 求实数$a$的取值范围;\\\\\n(2) 求当实数$a$为何值时, 以线段$AB$为直径的圆经过坐标原点.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "双曲线" ], "genre": "解答题", "ans": "", @@ -217037,7 +217694,8 @@ "content": "写出分别满足下列条件的抛物线的标准方程.\\\\\n(1) 焦点是$F(1,0)$;\\\\\n(2) 准线方程是$x=-2$.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "抛物线" ], "genre": "解答题", "ans": "", @@ -217060,7 +217718,8 @@ "content": "在抛物线$y^2=20x$上求一点$P$, 使点$P$与焦点的距离等于$15$.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "抛物线" ], "genre": "解答题", "ans": "", @@ -217081,7 +217740,8 @@ "content": "求抛物线$y^2=x$的一组斜率为$2$的平行弦的中点的轨迹方程.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "抛物线" ], "genre": "解答题", "ans": "", @@ -217102,7 +217762,8 @@ "content": "抛物线$y^2=2x$上的$AB$两点到焦点$F$的距离之和是$5$, 求线段$AB$的中点的横坐标.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "抛物线" ], "genre": "解答题", "ans": "", @@ -217125,7 +217786,8 @@ "content": "求抛物线$x=ay^2(a>0)$的焦点坐标与准线方程.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "抛物线" ], "genre": "解答题", "ans": "", @@ -217148,7 +217810,8 @@ "content": "过抛物线$y^2=2px(p>0)$的焦点的一条直线与抛物线相交于两个不同的点, 两个交点的纵坐标分别为$y_1,y_2$, 求证: $y_1y_2=-p^2$.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "抛物线" ], "genre": "解答题", "ans": "", @@ -217171,7 +217834,8 @@ "content": "抛物线$x^2=-32y$的焦点坐标是\\blank{50}, 准线方程是\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "抛物线" ], "genre": "填空题", "ans": "", @@ -217194,7 +217858,8 @@ "content": "已知抛物线的顶点在原点, 对称轴为$x$轴, 且过点$(-2,3)$, 求此抛物线的标准方程.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "抛物线" ], "genre": "解答题", "ans": "", @@ -217215,7 +217880,8 @@ "content": "已知抛物线$y^2=8x$的焦点为$F$, $P$在此抛物线上, 且$|PF|=5$, 求点$P$的坐标.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "抛物线" ], "genre": "解答题", "ans": "", @@ -217236,7 +217902,8 @@ "content": "已知一隧道的顶部是抛物拱形, 拱高是$1$米, 跨度为$2$米, 建立适当的直角坐标系, 求相应坐标系下此拱形的抛物线方程.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "抛物线" ], "genre": "解答题", "ans": "", @@ -217257,7 +217924,8 @@ "content": "已知直线$l$: $y=kx-4$与抛物线$y^2=8x$有且只有一个公共点, 求实数$k$的值.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "抛物线" ], "genre": "解答题", "ans": "", @@ -217282,7 +217950,8 @@ "content": "已知正三角形$ABC$的顶点$A$位于坐标原点, 顶点$B$与$C$均在抛物线$y^2=2x$上, 求$\\triangle ABC$的边长.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "抛物线" ], "genre": "解答题", "ans": "", @@ -217303,7 +217972,8 @@ "content": "已知直线$l$垂直于$x$轴, 且交抛物线$y^2=4x$于点$AB$, 且$|AB|=4\\sqrt 3$, 求直线$AB$的方程.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "抛物线" ], "genre": "解答题", "ans": "", @@ -217324,7 +217994,8 @@ "content": "在抛物线$x^2=\\dfrac 14y$上求一点$M$, 使点$M$到直线$y=4x-5$的距离最短.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "抛物线" ], "genre": "解答题", "ans": "", @@ -217345,7 +218016,8 @@ "content": "过点$Q(4,1)$作抛物线$y^2=8x$的弦$AB$, $AB$恰好被点$Q$平分, 求$AB$所在直线的方程.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "抛物线" ], "genre": "解答题", "ans": "", @@ -217366,7 +218038,8 @@ "content": "已知过抛物线$y^2=4x$的焦点$F$的直线交抛物线于$AB$两点, 过原点$O$作$\\overrightarrow {OM}$, 使$\\overrightarrow {OM}\\perp \\overrightarrow {AB}$, 垂足为$M$, 求点$M$的轨迹方程.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "抛物线" ], "genre": "解答题", "ans": "", @@ -217387,7 +218060,8 @@ "content": "抛物线$y^2=8x$的动弦$AB$的长为$16$, 求弦$AB$的中点$M$到$y$轴的最短距离.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "抛物线" ], "genre": "解答题", "ans": "", @@ -217408,7 +218082,8 @@ "content": "``太阳火''(sunfire)是一种利用太阳能的装置, 其截面是抛物线形状, 如图所示.``太阳火''依靠抛物线形状的镜面把反射的太阳光聚焦于抛物线的焦点处的锅炉, 加热产生的蒸汽推动汽轮发电机产生电能.根据图中所注尺寸, 求``太阳火''装置中截面抛物线的方程(其中抛物线截面深$10$英尺, 抛物线开口宽$37$英尺).\n\\begin{center}\n \\begin{tikzpicture}[scale = 0.15,>=latex]\n \\draw [domain = -18.5:18.5] plot (\\x,{\\x*\\x/40});\n \\filldraw [fill = gray!50, draw = black] (0,10) ellipse (2 and 1);\n \\draw (-2,10) -- (2,10);\n \\draw [->] (20,7) -- (20,10);\n \\draw [->] (20,3) -- (20,0);\n \\draw (19.5,0) -- (20.5,0) (19.5,10) -- (20.5,10);\n \\draw [dashed] (2,10) -- (19.5,10);\n \\draw (20,5) node {$10$英尺};\n \\draw [->] (-6,-2) -- (-18.5,-2);\n \\draw [->] (6,-2) -- (18.5,-2);\n \\draw (0,-2) node {$37$英尺};\n \\draw (-18.5,-1.5) -- (-18.5,-2.5) (18.5,-1.5) -- (18.5, -2.5);\n \\draw [dashed] (0,0) -- (0,10);\n \\draw (-2,10) node [left] {锅炉};\n \\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [ - "第七单元" + "第七单元", + "抛物线" ], "genre": "解答题", "ans": "", @@ -217584,7 +218259,8 @@ "content": "已知$F_1,F_2$为椭圆$\\dfrac{x^2}{16}+\\dfrac{y^2}9=1$的两个焦点, 过点$F_2$的直线交椭圆于$AB$两点, 且$|AB|=5$, 求$|AF_1|+|BF_1|$的值.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "椭圆" ], "genre": "解答题", "ans": "", @@ -217605,7 +218281,8 @@ "content": "已知倾斜角为$\\dfrac{\\pi }4$的直线交椭圆$\\dfrac{x^2}4+y^2=1$于$A,B$两点, 求线段$AB$的中点$P$的轨迹方程.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "椭圆" ], "genre": "解答题", "ans": "", @@ -217626,7 +218303,8 @@ "content": "已知过点$M(-2,0)$的直线$l$与椭圆$x^2+2y^2=2$交于$P_1,P_2$两点, 线段$P_1P_2$的中点为$P$, 设直线$l$的斜率为$k_1(k_1\\ne 0)$, 直线$OP$的斜率为$k_2$, 求证: $k_1\\cdot k_2$的值为定值.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "椭圆" ], "genre": "解答题", "ans": "", @@ -217647,7 +218325,8 @@ "content": "已知椭圆$\\dfrac{x^2}{45}+\\dfrac{y^2}{20}=1$的焦点分别是$F_1,F_2$, 过中心$O$作直线与椭圆相交于$A,B$两点, 若要使$\\triangle ABF_2$的面积是$20$, 求直线$AB$的方程.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "椭圆" ], "genre": "解答题", "ans": "", @@ -217668,7 +218347,8 @@ "content": "椭圆$x^2+4y^2=4$的长轴上的一个顶点为$A$, 以$A$为直角顶点作一个内接于此椭圆的等腰直角三角形, 求这个三角形的面积.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "椭圆" ], "genre": "解答题", "ans": "", @@ -217689,7 +218369,8 @@ "content": "已知点$P$是双曲线$\\dfrac{x^2}4-y^2=1$上任意一点, $O$为原点, 求线段$OP$的中点$Q$的轨迹方程.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "双曲线" ], "genre": "解答题", "ans": "", @@ -217710,7 +218391,8 @@ "content": "已知$F_1F_2$为双曲线$\\dfrac{x^2}{a^2}-\\dfrac{y^2}{b^2}=1(a>0,b>0)$的两个焦点, 过点$F_2$且垂直于$x$轴的直线交双曲线于点$P$, 且$\\angle F_1PF_2=60^{\\circ }$, 求此双曲线的渐近线的方程.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "双曲线" ], "genre": "解答题", "ans": "", @@ -217731,7 +218413,8 @@ "content": "过点$B(1,1)$能否作直线$l$, 使它与双曲线$x^2-\\dfrac{y^2}2=1$交于$Q_1,Q_2$两点, 且点$B$是线段$Q_1Q_2$的中点? 如果存在, 求此直线的方程; 如果不存在, 请说明理由.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "双曲线" ], "genre": "解答题", "ans": "", @@ -217752,7 +218435,8 @@ "content": "已知抛物线的焦点在$y$轴上, 抛物线上一点$M(a,-4)$到焦点$F$的距离为$5$, 求此抛物线的标准方程及实数$a$的值.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "抛物线" ], "genre": "解答题", "ans": "", @@ -217773,7 +218457,8 @@ "content": "过抛物线$y^2=4x$的焦点作直线交抛物线于$A(x_1,y_1)$、$B(x_2,y_2)$两点, 且$x_1+x_2=6$, 求$|AB|$的值.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "抛物线" ], "genre": "解答题", "ans": "", @@ -217794,7 +218479,8 @@ "content": "已知直线$y=x-2$与抛物线$y^2=ax$相交于$AB$两点, 且$OA\\perp OB$, 求实数$a$的值.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "抛物线" ], "genre": "解答题", "ans": "", @@ -217815,7 +218501,9 @@ "content": "已知抛物线的顶点是双曲线$16x^2-9y^2=144$的中心, 它的焦点是双曲线的左顶点, 求此抛物线的方程.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "双曲线", + "抛物线" ], "genre": "解答题", "ans": "", @@ -217836,7 +218524,8 @@ "content": "如图, 一位运动员在距篮下$4$米处跳起投篮, 球运行的路线是抛物线, 当球运行的水平距离为$2.5$米时, 达到最大高度为$3.5$米, 然后准确落入篮框, 已知篮框中心到地面的距离为$3.05$米.\\\\\n(1) 建立如图所示的平面直角坐标系, 求抛物线的方程;\\\\\n(2) 该运动员身高为$1.8$米, 在这次跳投中, 球在头顶上方$0.25$米处出手, 问球出手时, 他跳离地面的高度是多少.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "抛物线" ], "genre": "解答题", "ans": "", @@ -217878,7 +218567,9 @@ "content": "若椭圆$\\dfrac{x^2}4+\\dfrac{y^2}{a^2}=1$与双曲线$\\dfrac{x^2}{a^2}-\\dfrac{y^2}2=1$有相同的焦点, 则实数$a$为\\bracket{20}.\n\\fourch{1}{$-1$}{$\\pm 1$}{不确定}", "objs": [], "tags": [ - "第七单元" + "第七单元", + "椭圆", + "双曲线" ], "genre": "选择题", "ans": "", @@ -217901,7 +218592,8 @@ "content": "填若抛物线$y^2=2px(p>0)$上一点$M$到焦点的距离为$a(a>\\dfrac p2)$, 则点$M$到准线的距离为\\blank{50}, 点$M$的横坐标为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "抛物线" ], "genre": "填空题", "ans": "", @@ -217922,7 +218614,8 @@ "content": "若双曲线$x^2-y^2=1$的右支上有一点$P$到直线$y=x$的距离为$\\sqrt 2$, 则点$P$的坐标为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "双曲线" ], "genre": "填空题", "ans": "", @@ -217943,7 +218636,9 @@ "content": "命题: 椭圆$\\dfrac{x^2}{25}+\\dfrac{y^2}9=1$与双曲线$\\dfrac{x^2}{11}-\\dfrac{y^2}5=1$的焦距相等.试将此命题推广到一般情形, 使已知命题成为推广后命题的一个特例:\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "椭圆", + "双曲线" ], "genre": "填空题", "ans": "", @@ -217987,7 +218682,8 @@ "content": "求渐近线方程为$3x\\pm 4y=0$, 焦点为椭圆$\\dfrac{x^2}{10}+\\dfrac{y^2}5=1$的一对顶点的双曲线的方程.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "双曲线" ], "genre": "解答题", "ans": "", @@ -218010,7 +218706,8 @@ "content": "设$F_1,F_2$为椭圆$\\dfrac{x^2}9+\\dfrac{y^2}4=1$的两个焦点, $P$为椭圆上任意一点, 已知$P,F_1,F_2$是一个直角三角形的三个顶点, 且$|PF_1|>|PF_2|$, 求$\\dfrac{|PF_1|}{|PF_2|}$的值.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "椭圆" ], "genre": "解答题", "ans": "", @@ -218031,7 +218728,9 @@ "content": "已知双曲线的渐近线方程为$y=\\pm x$, 它的两个焦点都在抛物线$x^2=y+2$上, 求此双曲线的方程.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "双曲线", + "抛物线" ], "genre": "解答题", "ans": "", @@ -218073,7 +218772,8 @@ "content": "(1) 已知直线$l$: $4x-y-1=0$与抛物线$x^2=2y$交于$A(x_A,y_A)$、$B(x_B,y_B)$两点, 直线$l$与$x$轴相交于点$C(x_C,0)$, 求证: $\\dfrac 1{x_A}+\\dfrac 1{x_B}=\\dfrac 1{x_C}$;\\\\\n(2) 试将第(1)题中的命题加以推广, 使得第(1)题中的命题是推广后得到的命题的特例, 并证明推广后得到的命题正确.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "抛物线" ], "genre": "解答题", "ans": "", @@ -220356,7 +221056,8 @@ "content": "若直线$ax+2y+2=0$与直线$3x-y-2=0$平行, 则实数$a=$\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "填空题", "ans": "", @@ -220419,7 +221120,8 @@ "content": "若点$A(-2,3)$在抛物线$y^2=2px(p>0)$的准线上, 则实数$p$的值为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "抛物线" ], "genre": "填空题", "ans": "", @@ -220482,7 +221184,8 @@ "content": "已知$F_1,F_2$是定点, $|F_1F_2|=6$.若动点$M$满足$|MF_1|+|MF_2|=6$, 则动点$M$的轨迹是\\bracket{20}.\n\\fourch{直线}{线段}{圆}{椭圆}", "objs": [], "tags": [ - "第七单元" + "第七单元", + "椭圆" ], "genre": "选择题", "ans": "", @@ -220503,7 +221206,8 @@ "content": "经过点$P(4,-2)$的抛物线的标准方程是\\bracket{20}.\n\\fourch{$y^2=x$或$x^2=y$}{$y^2=x$或$x^2=8y$}{$x^2=y$或$y^2=-8x$}{$y^2=x$或$x^2=-8y$}", "objs": [], "tags": [ - "第七单元" + "第七单元", + "抛物线" ], "genre": "选择题", "ans": "", @@ -220545,7 +221249,8 @@ "content": "已知点$A(m,m+1)$与点$B(2,m-1)$, 求直线$AB$的倾斜角$\\theta$和斜率$k$.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -220566,7 +221271,8 @@ "content": "已知直线$mx+4y-2=0$与$2x-5y+n=0$互相垂直, 垂足为$(1,p)$, 求实数$m,n,p$的值.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -220587,7 +221293,8 @@ "content": "已知直线$l_1$通过点$(0,-3)$, 且方向向量为$\\overrightarrow d=(1,2)$, 直线$l_2$的方程是$3x+y+2=0$.求这两条直线的夹角$\\alpha$的大小.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -220608,7 +221315,8 @@ "content": "已知双曲线的中心是坐标原点, 它的一条渐近线方程为$3x-4y=0$, 且此双曲线经过点$(2,1)$, 求此双曲线的标准方程.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "双曲线" ], "genre": "解答题", "ans": "", @@ -220629,7 +221337,8 @@ "content": "已知椭圆$\\dfrac 12x^2+y^2=1$和椭圆外一点$(0,2)$, 过这点引直线与椭圆交于$A,B$两点, 求弦$AB$的中点$P$的轨迹方程.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "椭圆" ], "genre": "解答题", "ans": "", @@ -220650,7 +221359,8 @@ "content": "已知双曲线的中心在原点, 且它的一个焦点为$F(\\sqrt 7,0)$, 直线$y=x-1$与其相交于$MN$两点, 线段$MN$中点的横坐标为$-\\dfrac 23$, 求此双曲线的方程.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "双曲线" ], "genre": "解答题", "ans": "", @@ -220671,7 +221381,9 @@ "content": "直线$x-y+1=0$与椭圆$mx^2+ny^2=1(m,n>0)$相交于$AB$两点, 弦$AB$的中点的横坐标是$-\\dfrac 13$.求双曲线$\\dfrac{y^2}{m^2}-\\dfrac{x^2}{n^2}=1$的两条渐近线所夹锐角的大小.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "椭圆", + "双曲线" ], "genre": "解答题", "ans": "", @@ -220692,7 +221404,8 @@ "content": "如图, 双曲线$x^2-\\dfrac{y^2}4=1$的左、右两个焦点为$F_1,F_2$, 第二象限内的一点$P$在双曲线上, 且$\\angle F_1PF_2=\\dfrac{\\pi }3$.\n\\begin{center}\n \\begin{tikzpicture}[>=latex, scale = 0.5]\n \\draw [->] (-4,0) -- (4,0) node [below] {$x$};\n \\draw [->] (0,-4) -- (0,4) node [left] {$y$};\n \\draw (0,0) node [below left] {$O$};\n \\foreach \\i in {-3,-2,-1,1,2,3}\n {\n \\draw (\\i,0.1) -- (\\i,0) node [below] {$\\i$};\n \\draw (0.1,\\i) -- (0,\\i) node [left] {$\\i$};\n };\n \\filldraw ({-sqrt(5)},0) circle (0.05) node [above] {$F_1$} coordinate (F1);\n \\filldraw ({sqrt(5)},0) circle (0.05) node [above] {$F_2$} coordinate (F2);\n \\draw [domain = -3.5:3.5] plot ({sqrt(\\x*\\x/4+1)},\\x);\n \\draw [domain = -3.5:3.5] plot ({-sqrt(\\x*\\x/4+1)},\\x);\n \\draw ({-3/sqrt(5)},{4/sqrt(5)}) node [left] {$P$} coordinate (P);\n \\draw (F1) -- (P) -- (F2);\n \\end{tikzpicture}\n\\end{center}\n(1) 求$|PF_1|\\cdot|PF_2|$;\\\\\n(2) 求点$P$的坐标.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "双曲线" ], "genre": "解答题", "ans": "", @@ -220776,7 +221489,8 @@ "content": "已知$A(-2,-1)$、$B(2,5)$两点.若直线$3x+ay-6=0$过线段$AB$的中点, 则实数$a$的值等于\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "填空题", "ans": "", @@ -220841,7 +221555,8 @@ "content": "已知双曲线的中心为原点, 两条渐近线方程是$y=\\pm \\dfrac 23x$.若这条双曲线过点$M(\\dfrac 92,-1)$, 则这条双曲线的焦距为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "双曲线" ], "genre": "填空题", "ans": "", @@ -220862,7 +221577,8 @@ "content": "若抛物线$x^2=y$上的点到直线$y=2x+b$的最短距离为$\\sqrt 5$, 则实数$b=$\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "抛物线" ], "genre": "填空题", "ans": "", @@ -220883,7 +221599,8 @@ "content": "若$\\theta \\in \\mathbf{R}$, 则方程$x^2+y^2\\sin \\theta =1$所表示的曲线一定不是\\bracket{20}.\n\\fourch{直线}{圆}{抛物线}{双曲线}", "objs": [], "tags": [ - "第七单元" + "第七单元", + "抛物线" ], "genre": "选择题", "ans": "", @@ -220946,7 +221663,8 @@ "content": "已知双曲线的中心在原点, 且它的一个焦点为$F_1(-\\sqrt 5,0)$.若点$P$位于此双曲线上, 线段$PF_1$的中点坐标为$(0,2)$, 则此双曲线的方程是\\bracket{20}.\n\\fourch{$\\dfrac{x^2}4-\\dfrac{y^2}1=1$}{$x^2-\\dfrac{y^2}4=1$}{$\\dfrac{x^2}2-\\dfrac{y^2}3=1$}{$\\dfrac{x^2}3-\\dfrac{y^2}2=1$}", "objs": [], "tags": [ - "第七单元" + "第七单元", + "双曲线" ], "genre": "选择题", "ans": "", @@ -220988,7 +221706,8 @@ "content": "在直线$x+3y=0$上求一点, 使它到原点的距离与它到直线$x+3y-2=0$的距离相等.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -221009,7 +221728,9 @@ "content": "已知抛物线$y^2=4x$与椭圆$\\dfrac{x^2}9+\\dfrac{y^2}k=1$有公共焦点$F_1$, 椭圆的另一焦点为$F_2$, $P$是这两条曲线的一个交点, 求$\\triangle PF_1F_2$的周长.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "椭圆", + "抛物线" ], "genre": "解答题", "ans": "", @@ -221030,7 +221751,8 @@ "content": "已知抛物线$y=2x^2$上有$A(x_1,y_2)$、$B(x_2,y_2)$两点, 且$,B$关于直线$y=x+m$对称, $x_1x_2=-\\dfrac 12$, 求实数$m$的值.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "抛物线" ], "genre": "解答题", "ans": "", @@ -221093,7 +221815,8 @@ "content": "已知直线$l$过点$A(-3,1)$, 且倾斜角为$45^{\\circ }$.直线$l$与焦点为$(-\\sqrt 6,0)$、$(\\sqrt 6,0)$的椭圆交于$B,C$两点, 且点$A$为线段$BC$的中点.是否存在满足上述条件的椭圆? 若存在, 求椭圆的方程; 若不存在, 请说明理由.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "椭圆" ], "genre": "解答题", "ans": "", @@ -221114,7 +221837,8 @@ "content": "如图, 直线$y=\\dfrac 12x$与抛物线$y=\\dfrac 18x^2-4$交于$AB$两点, 线段$AB$的垂直平分线与直线$y=-5$交于点$Q$.\n\\begin{center}\n \\begin{tikzpicture}[>=latex,scale = 0.2]\n \\draw [->] (-10,0) -- (10,0) node [below] {$x$};\n \\draw [->] (0,-10) -- (0,10) node [left] {$y$};\n \\draw (0,0) node [above left] {$O$};\n \\draw [name path = line, domain = -8:10] plot (\\x,{\\x/2});\n \\draw [name path = para, domain = -9:9] plot (\\x,{pow(\\x,2)/8-4});\n \\draw (-10,-5) -- (10,-5);\n \\draw (0,-5) node [below left] {$-5$};\n \\draw (-4,-2) node [below] {$A$} coordinate (A);\n \\draw (8,4) node [below right] {$B$} coordinate (B);\n \\filldraw (6,{36/8-4}) circle (0.1) node [right] {$P$};\n \\draw (-2,9) coordinate (T) -- (6,-7) (5,-5) node [below left] {$Q$};\n \\draw (2,1) coordinate (V);\n \\draw (V) pic [draw, scale = 0.3] {right angle = B--V--T};\n \\end{tikzpicture}\n\\end{center}\n(1) 求点$Q$的坐标;\\\\\n(2) 当$P$为抛物线上位于线段$AB$下方(含点$AB$)的动点时, 求$\\triangle OPQ$面积的最大值.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "抛物线" ], "genre": "解答题", "ans": "", @@ -236186,7 +236910,8 @@ "content": "求经过下列两点的直线的斜率和倾斜角:\\\\\n(1) $P(-2, 2)$、$Q(2, -2)$;\\\\\n(2) $P(5,\\sqrt 3)$、$Q(2,2\\sqrt 3)$.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -236207,7 +236932,8 @@ "content": "在平面直角坐标系中有一个边长为$1$的正方形$OABC$, 其中$O$为坐标原点, 点$A$、$C$分别在$x$轴和$y$轴上, 点$B$在第一象限. 求直线$OB$和$AC$的斜率.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -236228,7 +236954,8 @@ "content": "证明: 在平面直角坐标系中, 如果两条直线平行, 那么它们的倾斜角相等.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -236249,7 +236976,8 @@ "content": "求经过点$P(-2,3)$且斜率为$-1$的直线$l$的点斜式方程.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -236270,7 +236998,8 @@ "content": "求倾斜角是$\\dfrac{5\\pi} 6$且在$x$轴上的截距为$-1$的直线$l$的点斜式方程.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -236291,7 +237020,8 @@ "content": "求经过点$A(2,3)$且垂直于$x$轴的直线$l$的方程.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -236312,7 +237042,8 @@ "content": "已知直线$l$经过点$M(-2,-1)$且在$x$轴、$y$轴上截距相等, 求$l$的方程.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -236333,7 +237064,8 @@ "content": "求经过点$A(-2,3)$、$B(0,6)$的直线$l$的两点式方程.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -236354,7 +237086,8 @@ "content": "已知三个不同的点$A(3,1)$、$B(a+1,3)$、$C(2a-1,3-a)$都在一条直线$l$上, 求实数$a$的值和直线$l$的方程.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -236375,7 +237108,8 @@ "content": "在平面直角坐标系中, $O$是坐标原点. 已知$A$、$B$两点的坐标分别为$(4,0)$、$\n(0,3)$, 分别求$\\triangle ABO$的三条边上的中线所在直线的方程.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -236396,7 +237130,8 @@ "content": "求下列方程所表示直线的斜率与倾斜角:\\\\\n(1) $x=1$;\\\\\n(2) $x+y-1=0$;\\\\\n(3) $x+2y-1=0$;\\\\\n(4) $y=1$.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -236417,7 +237152,8 @@ "content": "求证: 无论实数$m$取何值, 直线$l:x+(m+1)y+1=0$都经过一个定点.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -236438,7 +237174,8 @@ "content": "已知直线$l:kx+2y+3-k=0$经过平面直角坐标系的第二、第三与第四象限, 求实数$k$的取值范围.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -236459,7 +237196,8 @@ "content": "写出下列直线的一个法向量:\\\\\n(1) $2x-3y+1=0$;\\\\\n(2) $3x+2y+1=0$;\\\\\n(3) $x+3=0$;\\\\\n(4) $y=\\dfrac 12x-3$.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -236480,7 +237218,8 @@ "content": "已知直线$l$的方程是$(a-3)x+(2a+1)y-3=0$, 它的一个法向量是$\\overrightarrow n=(3,2)$. 求实数$a$的值.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -236501,7 +237240,8 @@ "content": "根据下列条件, 求直线$l$的方程:\\\\\n(1) $l$在$x$轴上的截距为$-1$, 且$l$的一个法向量是$\\overrightarrow n=(-1,2)$;\\\\\n(2) $l$经过点$(2,3)$, 且$l$上的任何向量都与向量$\\overrightarrow a=(1,2)$平行.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -236522,7 +237262,8 @@ "content": "判断下列两条直线的位置关系. 若相交, 求交点坐标.\\\\\n(1) $l_1: x+3y+1=0$, $l_2: 3x+4=0$;\\\\\n(2)$ l_1: x-3y+1=0$, $l_2: y=\\dfrac 13x+4$.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -236543,7 +237284,8 @@ "content": "已知直线$l_1: (a+1)x+y+a=0$, $l_2:x+(a+1)y-2=0$. 若$l_1\\parallel l_2$, 求实数$a$的值.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -236564,7 +237306,8 @@ "content": "求经过直线$l_1: x-y-4=0$与$l_2:2x-3y-7=0$的交点, 且与直线$l_3: 2x+y+1=0$平行的直线$l$的方程.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -236585,7 +237328,8 @@ "content": "已知直线$l_1:(a-2)x+ay-2=0$与$l_2:(1-a)x+(a+1)y+1=0$互相垂直, 求实数$a$的值.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -236606,7 +237350,8 @@ "content": "求过点$(-1,-1)$且分别与下列直线垂直的直线方程:\\\\\n(1) $y=2$;\\\\\n(2) $y=x$;\\\\\n(3) $2x+y+2=0$;\\\\\n(4) $x\\cos\\theta +y\\sin \\theta =1$, $\\theta$为给定的实数.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -236627,7 +237372,8 @@ "content": "根据下列方程, 求直线$l_1$与$l_2$的夹角的大小:\\\\\n(1) $l_1: 3x-5y+1=0$与$l_2:2x+y=3$;\\\\\n(2) $l_1: y=5x-3$与$l_2:y=-3x+2$.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -236648,7 +237394,8 @@ "content": "已知直线$l1:\\sqrt 3x-y+3=0$与直线$l_2: y=kx+3$的夹角为$\\dfrac \\pi 4$, 求实数$k$的值.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -236669,7 +237416,8 @@ "content": "求经过点$A(4,-3)$且与直线$l: x+y-3=0$的夹角为$\\dfrac\\pi 3$的直线$l'$的方程.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -236690,7 +237438,8 @@ "content": "根据下列条件, 求点$M(-2,-1)$到直线$l$的距离$d$:\\\\\n(1) $l: x=3$;\\\\\n(2) $l: y=3$;\\\\\n(3) $l: x+y=3$;\\\\\n(4) $l: y=3x-5$.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -236711,7 +237460,8 @@ "content": "在直角三角形$ABC$中, $\\angle A=\\dfrac \\pi 2$, $|AB|=6$, $|AC|=8$. 求三角形的重心$G$到斜边$BC$所在直线的距离.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -237010,7 +237760,8 @@ "content": "分别写出满足下列条件的椭圆的标准方程:\\\\\n(1) 焦点在$y$轴上, 焦距为$2\\sqrt {15}$, 且经过点$(0,-4)$;\\\\\n(2) 焦距为$4$, 且经过点$(\\sqrt 5,0)$.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "椭圆" ], "genre": "解答题", "ans": "", @@ -237031,7 +237782,8 @@ "content": "已知下列椭圆的方程, 分别求椭圆的长轴长、短轴长、焦点坐标和顶点坐标:\\\\\n(1) $\\dfrac{x^2}4+\\dfrac{y^2}3=1$;\\\\\n(2) $25x^2+4y^2=100$.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "椭圆" ], "genre": "解答题", "ans": "", @@ -237052,7 +237804,8 @@ "content": "用离心率作为指标衡量, 下列每组两个椭圆中哪一个更接近圆?\\\\\n(1) $\\dfrac{x^2}4+\\dfrac{y^2}9=1$与$\\dfrac{x^2}{16}+\\dfrac{y^2}{12}=1$;\\\\\n(2) $x^2+9y^2=36$与$5x^2+3y^2=30$.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "椭圆" ], "genre": "解答题", "ans": "", @@ -237073,7 +237826,8 @@ "content": "若一椭圆以原点为中心, 一个焦点的坐标为$(\\sqrt 2,0)$, 且长轴长是短轴长的$\\sqrt 3$倍. 求该椭圆的标准方程.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "椭圆" ], "genre": "解答题", "ans": "", @@ -237094,7 +237848,8 @@ "content": "已知$P$是椭圆$\\dfrac{x^2}{36}+\\dfrac{y^2}{20}=1$上一个动点, $F_1$是椭圆的左焦点. 求$|PF_1|$的最大值和最小值.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "椭圆" ], "genre": "解答题", "ans": "", @@ -237115,7 +237870,8 @@ "content": "点$P$在焦点为$F_1$、$F_2$的椭圆$\\dfrac{x^2}{45}+\\dfrac{y^2}{20}=1$上, 且$\\angle F_1PF_2=90^\\circ$. 求$|PF1|\\cdot |PF_2|$的值.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "椭圆" ], "genre": "解答题", "ans": "", @@ -237136,7 +237892,8 @@ "content": "已知直线$l: y=mx-2$与椭圆$C: \\dfrac{x^2}4+\\dfrac{y^2}3=1$相交于两个不同的点, 求实数$m$的取值范围.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "椭圆" ], "genre": "解答题", "ans": "", @@ -237178,7 +237935,8 @@ "content": "已知双曲线$\\dfrac{x^2}9-\\dfrac{y^2}m=1$的焦点在$x$轴上, 焦距为$10$. 求实数$m$的值.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "双曲线" ], "genre": "解答题", "ans": "", @@ -237199,7 +237957,8 @@ "content": "已知双曲线$\\dfrac{x^2}{16}-\\dfrac{y^2}9=1$的两个焦点分别为$F_1$、$F_2$, $P$为双曲线上一点, 且$\\angle F_1PF_2=\\dfrac \\pi 2$. 求$\\triangle PF_1F_2$的面积.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "双曲线" ], "genre": "解答题", "ans": "", @@ -237220,7 +237979,8 @@ "content": "分别写出下列双曲线的实半轴长、虚半轴长、离心率、焦点坐标、顶点坐标和渐近线方程:\\\\\n(1) $9x^2-16y^2=144$;\\\\\n(2) $\\dfrac{y^2}4-\\dfrac{x^2}3=1$.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "双曲线" ], "genre": "解答题", "ans": "", @@ -237241,7 +238001,8 @@ "content": "在下列双曲线中, 以$y=\\pm \\dfrac 12x$为渐近线的是\n\\bracket{20}.\n\\fourch{$\\dfrac{x^2}{16}-\\dfrac{y^2}4=1$}{$\\dfrac{x^2}4-\\dfrac{y^2}{16}=1$}{$\\dfrac{x^2}2-y^2=1$}{$x^2-\n\\dfrac{y^2}2=1$}", "objs": [], "tags": [ - "第七单元" + "第七单元", + "双曲线" ], "genre": "选择题", "ans": "", @@ -237264,7 +238025,8 @@ "content": "判断双曲线$\\dfrac{x^2}4-\\dfrac{y^2}5=1$与双曲线$\\dfrac{y^2}5-\\dfrac{x^2}4=1$的四个焦点是否共圆.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "双曲线" ], "genre": "解答题", "ans": "", @@ -237285,7 +238047,8 @@ "content": "求适合下列条件的双曲线的标准方程:\\\\\n(1) 顶点在$x$轴上, 两顶点间的距离是$10$, 且经过点$(10,3)$;\\\\\n(2) 一个焦点的坐标为$(5,0)$, 一条渐近线方程为$3x-4y=0$.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "双曲线" ], "genre": "解答题", "ans": "", @@ -237306,7 +238069,8 @@ "content": "给定一对直线$y=\\pm\\dfrac ba x$($a>0$, $b>0$), 写出所有以这对直线为渐近线的、实轴在$x$轴上的双曲线的方程.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "双曲线" ], "genre": "解答题", "ans": "", @@ -237327,7 +238091,8 @@ "content": "联系双曲线$\\dfrac{x^2}{a^2}-\\dfrac{y^2}{b^2}=1$($a>0$, $b>0$)的性质, 讨论并叙述双曲线$\\dfrac{y^2}{a^2}-\\dfrac{x^2}{b^2}=1$($a>0$, $b>0$)的性质(不要求推理过程).", "objs": [], "tags": [ - "第七单元" + "第七单元", + "双曲线" ], "genre": "解答题", "ans": "", @@ -237369,7 +238134,8 @@ "content": "分别写出满足下列条件的抛物线的标准方程:\\\\\n(1) 焦点是$F(-2,0)$;\\\\\n(2) 准线方程是$y=1$.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "抛物线" ], "genre": "解答题", "ans": "", @@ -237392,7 +238158,8 @@ "content": "求抛物线$y^2=4x$上到焦点的距离等于$9$的点的坐标.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "抛物线" ], "genre": "解答题", "ans": "", @@ -237413,7 +238180,8 @@ "content": "过点$P(2,4)$且与抛物线$y2=8x$有且只有一个公共点的直线有\n\\bracket{20}.\n\\fourch{$1$条}{$2$条}{$3$条}{$4$条}", "objs": [], "tags": [ - "第七单元" + "第七单元", + "抛物线" ], "genre": "选择题", "ans": "", @@ -237434,7 +238202,8 @@ "content": "求抛物线$y^2=4x$上的点到直线$4x+3y+7=0$的最短距离.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "抛物线" ], "genre": "解答题", "ans": "", @@ -237455,7 +238224,8 @@ "content": "由抛物线的标准方程知, 函数$y=\\sqrt x$的图像是某条抛物线的一部分. 求这条抛物线的焦点坐标和准线方程.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "抛物线" ], "genre": "解答题", "ans": "", @@ -237581,7 +238351,8 @@ "content": "动点$M$作匀速直线运动, 它在$x$轴和$y$轴方向的分速度分别为$9$和$12$, 运动开始时, 点$M$位于$A(1,1)$. 求点$M$的轨迹的参数方程.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -237644,7 +238415,8 @@ "content": "求经过点$A(a,0)$且和极轴垂直的直线$l$的极坐标方程.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -240724,7 +241496,8 @@ "content": "双曲线$\\dfrac{x^2}{9}-y^2=1$的实轴长为\\blank{50}.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "双曲线" ], "genre": "填空题", "ans": "", @@ -241133,7 +241906,8 @@ "content": "已知椭圆方程$\\Gamma: \\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1$($a>b>0$)的左、右焦点为$F_1(-\\sqrt{2},0)$、$F_2(\\sqrt{2},0)$, $A$为椭圆的下顶点, $M$为直线$l:x+y-4\\sqrt{2}=0$上一点.\\\\\n(1) 若$a=2$, $AM$的中点在$x$轴上, 求点$M$的坐标;\\\\\n(2) 直线$l$交$y$轴于点$B$, 直线$AM$经过$F_2$, 若$\\triangle ABM$有一个内角的余弦值为$\\dfrac 35$, 求$b$的值;\\\\\n(3) 若$\\Gamma$上存在点$P$到直线$l$的距离为$d$, 且满足$d+|PF_1|+|PF_2|=6$, 当$a$变化时, 求$d$的最小值.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "椭圆" ], "genre": "解答题", "ans": "", @@ -254796,7 +255570,8 @@ "content": "如图, 在平面直角坐标系中, 直线$l_1$与$l_2$垂直, 垂足为$A$, $l_1$、$l_2$与$x$轴的交点分别为$B$、$C$, $\\angle ABC=\\dfrac \\pi 6$. 试分别求直线$l_1$、$l_2$的倾斜角和斜率.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [->, name path = x] (-2,0) -- (2,0) node [below] {$x$};\n\\draw [->, name path = y] (0,-2) -- (0,2) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [name path = l1] (-1.9,0.2) --++ (-30:3) node [right] {$l_1$};\n\\draw [name path = l2] (-0.4,-1.8) --++ (60:3) node [right] {$l_2$};\n\\draw [name intersections = {of = l1 and l2, by = A}];\n\\draw [name intersections = {of = l1 and x, by = B}];\n\\draw [name intersections = {of = l2 and x, by = C}];\n\\draw (B) node [below] {$B$};\n\\draw (C) node [above left] {$C$};\n\\draw (A) node [right] {$A$};\n\\draw (A) pic [draw, scale = 0.3] {right angle = C--A--B};\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -254817,7 +255592,8 @@ "content": "求经过下列两点的直线的斜率与倾斜角:\\\\\n(1) $P(1, 2)$、$Q(2, -1)$;\\\\\n(2) $M(2, 1)$、$N(a, -2)$, 其中实数$a$是常数.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -254838,7 +255614,8 @@ "content": "根据下列直线$l$的倾斜角$\\theta$的取值范围, 计算斜率$k$的取值范围:\\\\\n(1) $\\theta \\in [\\dfrac \\pi 4, \\dfrac \\pi 3]$;\\\\\n(2) $\\theta \\in (\\dfrac \\pi 2, \\dfrac{2\\pi} 3)$.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -254859,7 +255636,8 @@ "content": "已知三个不同的点$A(2, a)$、$B(a+1, 2a+1)$、$C(-4, 1+a)$在同一条直线上, 求实数$a$的值及该直线的斜率.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -254880,7 +255658,8 @@ "content": "如图, 已知点$A(2, 4)$、$B(-1, -1)$、$C(4, 1)$, 过点$B$的直线$l$与线段$AC$相交. 求直线$l$的斜率$k$的取值范围.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.4]\n\\draw [->] (-2,0) -- (6,0) node [below] {$x$};\n\\draw [->] (0,-2) -- (0,6) node [left] {$y$};\n\\draw (0,0) node [below right] {$O$};\n\\filldraw (2,4) circle (0.1) node [left] {$A$} coordinate (A);\n\\filldraw (-1,-1) circle (0.1) node [left] {$B$} coordinate (B);\n\\filldraw (4,1) circle (0.1) node [right] {$C$} coordinate (C);\n\\draw (A) -- (C);\n\\draw (B) ++ (-1,-0.9) --++ (7,6.3) node [right] {$l$};\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -254901,7 +255680,8 @@ "content": "已知常数$\\theta \\in [0, \\pi)$, 试用$\\theta$表示经过$P(0, 0)$、$Q(\\sin \\theta , \\cos \\theta)$两点的直线$l$的倾斜角.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -254922,7 +255702,8 @@ "content": "设直线$l_1$、$l_2$的倾斜角分别为$\\theta_1$、$\\theta_2$, 求证: $l_1\\perp l_2$的充要条件是$|\\theta_1-\\theta_2|=\\dfrac \\pi 2$.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -254943,7 +255724,8 @@ "content": "已知直线$l$在平面直角坐标系中的斜率是$k$, 向量$\\overrightarrow a$在直线$l$上. 求向量$\\overrightarrow a$在$x$轴上的投影向量.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -254964,7 +255746,8 @@ "content": "已知直线$l$经过点$P(3, 5)$, 倾斜角为$\\alpha$且$\\cos\\alpha=\\dfrac 35$. 求直线$l$的点斜式方程.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -254985,7 +255768,8 @@ "content": "已知直线$l$在$y$轴上的截距为$4$, 倾斜角为$\\alpha$且$\\sin\\alpha=\\dfrac 35$. 求直线$l$的斜截式方程.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -255006,7 +255790,8 @@ "content": "求下列直线的斜率与在$x$、$y$两坐标轴上的截距:\\\\\n(1) $l_1: y+1=-\\dfrac {\\sqrt 3}3(x+1)$;\\\\\n(2) $l_2: y=-3x+\\sqrt 3$.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -255027,7 +255812,8 @@ "content": "已知直线$l: y=kx+2$经过点$(1, -3)$.\\\\\n(1) 求$l$的倾斜角的大小;\\\\\n(2) 求$l$在$x$轴上的截距.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -255048,7 +255834,8 @@ "content": "直线$l$经过点$P(-2, 1)$, 在$x$轴、$y$轴上的截距分别为$a$、$b$. 已知$a+b=4$, 求直线$l$的方程.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -255069,7 +255856,8 @@ "content": "根据给定条件, 求下列直线的两点式方程:\\\\\n(1) 直线$l_1$经过点$A(2, 0)$、$B(3, 7)$;\\\\\n(2) 直线$l_2$与坐标轴的交点分别为$(3, 0)$、$(0, -1)$.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -255090,7 +255878,8 @@ "content": "已知$\\triangle ABC$的三个顶点的坐标分别为$A(3, 8)$、$B(3, -2)$、$C(-3, 0)$.\\\\\n(1)求边$BC$所在直线的方程;\\\\\n(2)求边$BC$上的中线所在直线的方程.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -255113,7 +255902,8 @@ "content": "设直线$l$在$x$轴与$y$轴上的截距分别是$a$与$b$, 且$a$与$b$均不为零. 求证: 直线$l$的方程可以写成$\\dfrac xa+\\dfrac yb=1$.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -255134,7 +255924,8 @@ "content": "一个弹簧在弹性限度内挂$4\\text{kg}$的物体时弹簧长度为$20\\text{cm}$, 挂$5\\text{kg}$物体时弹簧长度为$21.5\\text{cm}$. 已知在弹性限度内所挂物体的质量$x$(单位: $\\text{kg}$)与弹簧长度$y$(单位: $\\text{cm}$)的关系可以用直线的方程表示, 求该直线的方程, 并求弹簧自身的长度.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -255155,7 +255946,8 @@ "content": "在平面直角坐标系中, 作出下列直线, 并求它们的斜率与倾斜角.\\\\\n(1) $l_1: 3x-y-2=0$;\\\\\n(2) $l_2: 3x+2y-1=0$.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -255176,7 +255968,8 @@ "content": "设直线$l$的方程是$ax+by+c=0$, 在下列条件下, 求实数$a$、$b$、$c$满足的条件:\\\\\n(1) $l$与$x$轴、$y$轴均相交;\\\\\n(2) $l$经过第二、第三、第四象限.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -255197,7 +255990,8 @@ "content": "已知直线$l: ax+(4-2a)y-3=0$, 根据下列条件, 求实数$a$的值:\\\\\n(1) $l$经过点$(1, 1)$;\\\\\n(2) $l$在两个坐标轴上的截距相等.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -255241,7 +256035,8 @@ "content": "已知直线$l_1: 3kx+(k+2)y+6=0$, 直线$l_2: kx+(2k-3)y+2=0$. 若这两条直线的法向量互相垂直, 求$k$的值.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -255262,7 +256057,8 @@ "content": "已知平行四边形$ABCD$中, 三个顶点的坐标分别为$A(1, 2)$、$B(3, 4)$、$C(2, 6)$.分别求边$AD$、$CD$所在直线的方程.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -255283,7 +256079,8 @@ "content": "已知直线$l$经过点$P(2, -1)$, 与$x$轴、$y$轴分别交于$A$、$B$两点. 若$2\\overrightarrow{PA}+\\overrightarrow{PB}=\\overrightarrow 0$, 求直线$l$的方程.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -255304,7 +256101,8 @@ "content": "直线$l: y=kx+b$($k$、$b\\in \\mathbf{R}$)与线段$AB$相交, 其中点$A$为$(4, 2)$, 点$B$为$(1, 5)$.\\\\\n(1) 当$b=-1$时, 求$k$的取值范围;\\\\\n(2) 当$k=1$时, 求$b$的取值范围.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -255325,7 +256123,8 @@ "content": "已知$\\triangle ABC$中, 两个顶点的坐标分别为$A(-2, 1)$、$B(4, -3)$, 点$G(0, 2)$是此三角形的重心. 求边$BC$、$AC$所在直线的方程.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -255346,7 +256145,8 @@ "content": "若$2x_1+3y_1=1, 2x_2+3y_2=1$, 且$x_1\\ne x_2$. 求经过两点$A(x_1, y_1)$、$B(x_2, y_2)$的直线$l$的方程.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -255367,7 +256167,8 @@ "content": "如图是一个$\\text{W}$形的霓虹灯(灯管宽度忽略不计), 每边长都是$2\\text{m}$, 每相邻两边相交所成的锐角都是$30^\\circ$. 试建立适当的平面直角坐标系, 写出此霓虹灯的每条边所在直线在这个坐标系中的方程.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [very thick] (0,0) --++ (-75:2) --++ (75:2) --++ (-75:2) --++ (75:2);\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -255388,7 +256189,8 @@ "content": "证明: 直线$2x+(1-\\cos 2\\theta)y-\\sin \\theta =0$($\\theta \\in \\mathbf{R}$且不是$\\pi$的整数倍)和两坐标轴围成图形的面积是定值.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -255409,7 +256211,8 @@ "content": "已知直线$l: (a-1)x+(3-2a)y+a+1=0$.\\\\\n(1) 若直线的斜率$k\\in [-1, 2]$, 求实数$a$的取值范围;\\\\\n(2) 求证:对任意实数$a$, 直线$l$都经过一个定点.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -255430,7 +256233,8 @@ "content": "根据下列方程, 判定直线$l_1$与$l_2$的位置关系:\\\\\n(1)$l_1: 2x-3y-1=0$, $l_2: 4x-6y-2=0$;\\\\\n(2)$l_1: y=\\dfrac 13x+1$, $l_2: x-6y-2=0$;\\\\\n(3)$l_1: (\\sqrt 5-1)x-2y+1=0$, $l_2: 2x-(\\sqrt 5-1)y-2=0$.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -255451,7 +256255,8 @@ "content": "已知直线$l_1: 6x+(t-1)y-8=0$, 直线$l_2: (t+4)$x+$(t+6)y-16=0$. 根据下列条件, 求实数$t$的取值范围:\\\\\n(1) $l_1$与$l_2$相交;\\\\\n(2) $l_1\\parallel l_2$;\\\\\n(3) $l_1$与$l_2$重合.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -255472,7 +256277,8 @@ "content": "已知两条直线$l_1: (t-1)x+2y-t=0$和$l_2: $$x+ty+t-2=0$, 且$l_1\\parallel l_2$. 求实数$t$的值.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -255493,7 +256299,8 @@ "content": "已知平行四边形$ABCD$中, 一组对边$AB$、$CD$所在直线的方程分别为$ax+4y=a+2$, $x+ay=a$. 求实数$a$的值.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -255535,7 +256342,8 @@ "content": "已知直线$l_1: (k-3)x+(5-k)y+1=0$与直线$l_2: 2(k-3)x-2y+(2-k)=0$互相垂直, 求实数$k$的值.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -255556,7 +256364,8 @@ "content": "已知直线$l$垂直于直线$l': 2x+3y-4=0$, 根据下列条件求$l$的方程:\\\\\n(1) $l$经过点$(1, 1)$;\\\\\n(2) $l$与坐标轴围成的三角形的面积是$3$.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -255577,7 +256386,8 @@ "content": "已知等腰直角三角形$ABC$的斜边$AB$所在直线的方程为$3x-y-5=0$, 直角顶点为$C(4, -1)$. 求两条直角边所在直线的方程.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -255600,7 +256410,8 @@ "content": "根据下列方程, 求直线$l_1$与$l_2$的夹角的大小:\\\\\n(1) $l_1: x+3y+2=0$, $l_2: 4x+2y-1=0$;\\\\\n(2) $l_1: x+2y-3=0$, $l_2: x-y-5=0$;\\\\\n(3) $l_1: 2x-3y+6=0$, $l_2: x-5=0$.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -255621,7 +256432,8 @@ "content": "若直线$x+my+5=0$与直线$x+y+1=0$的夹角为$\\pi 4$, 求实数$m$的值.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -255642,7 +256454,8 @@ "content": "已知等腰直角三角形$ABC$的直角边$BC$所在直线的方程为$x-2y-6=0$, 顶点$A$的坐标为$(0, 6)$. 分别求直角边$AC$、斜边$AB$所在直线的方程.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -255728,7 +256541,8 @@ "content": "分别求经过直线$l_1: 5x+2y-3=0$和$l_2: $$3x-5y-8=0$的交点, 且与直线$x+4y-7=0$垂直、平行的直线的方程.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -255749,7 +256563,8 @@ "content": "已知$\\triangle ABC$的一个顶点为$A(3, -4)$, 有两条高所在直线的方程分别是$7x-2y-1=0$与$2x-7y-6=0$. 求$\\triangle ABC$三条边所在直线的方程.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -255770,7 +256585,8 @@ "content": "求直线$l_1: x+y-3=0$与直线$l_2: 7x-y-5=0$夹角平分线的方程.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -255791,7 +256607,8 @@ "content": "一束光线经过点$(-2, 1)$, 由直线$l: y=x$反射后, 经过点$(3, 5)$射出. 求反射光线所在直线的方程.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -255812,7 +256629,8 @@ "content": "求点$P(2, 3)$到直线$l$的距离:\\\\\n(1) $l: 3x-2y=13$;\\\\\n(2) $l: y=-2x+3$.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -255833,7 +256651,8 @@ "content": "已知点$A(a, 6)$到直线$3x-4y-4=0$的距离等于$4$, 求实数$a$的值.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -255875,7 +256694,8 @@ "content": "已知直线$l_1: 2x-y+a=0$与直线$l_2: -4x+2y+1=0$的距离为$\\dfrac{7\\sqrt 5}{10}$, 求实数$a$的值.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -255898,7 +256718,8 @@ "content": "已知点$A(1, 0)$、$B(4, -4)$. 若点$A$与点$B$到直线$l$的距离都为$2$, 求直线$l$的方程.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -255919,7 +256740,8 @@ "content": "已知点$P$是直线$3x-4y+2=0$上任意一点, 求点$P$与点$A(3, -1)$之间距离的最小值.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -255940,7 +256762,8 @@ "content": "已知直线$l$经过点$P(1, 1)$且与直线$l_1: y=\\sqrt 3x+1$和$l_2: y=\\sqrt 3x+3$分别交于点$A$和点$B$. 若$|AB|=\\sqrt 2$, 求直线$l$的方程.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -256427,7 +257250,8 @@ "content": "若方程$16x^2+ky^2=16k$表示焦点在$y$轴上的椭圆, 求实数$k$的取值范围.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "椭圆" ], "genre": "解答题", "ans": "", @@ -256448,7 +257272,8 @@ "content": "设$F$是椭圆的一个焦点, $B_1B_2$是椭圆的短轴, $\\angle B_1FB_2=60^\\circ$. 求椭圆的离心率.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "椭圆" ], "genre": "解答题", "ans": "", @@ -256469,7 +257294,8 @@ "content": "已知椭圆的一个焦点是$F_1(-3, 0)$, 且经过点$P(2, \\sqrt 2)$. 求这个椭圆的标准方程.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "椭圆" ], "genre": "解答题", "ans": "", @@ -256490,7 +257316,8 @@ "content": "直线$y=2x+b$被椭圆$4x^2+y^2=16$所截得的弦长为$\\sqrt{35}$, 求实数$b$的值.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "椭圆" ], "genre": "解答题", "ans": "", @@ -256511,7 +257338,8 @@ "content": "若对于任意实数$k$, 直线$y=kx+1$与椭圆$\\dfrac{x^2}5+\\dfrac{y^2}m=1$恒有公共点. 求实数$m$的取值范围.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "椭圆" ], "genre": "解答题", "ans": "", @@ -256532,7 +257360,8 @@ "content": "已知$P$是椭圆$\\dfrac{x^2}{25}+\\dfrac{y^2}9=1$上的点, $F_1$、$F_2$是椭圆的两个焦点.\\\\\n(1) 若$\\angle F_1PF_2=60^\\circ$, 求$\\triangle PF_1F_2$的面积;\\\\\n(2) 若$\\triangle PF_1F_2$的面积为$9$, 求$\\angle F_1PF_2$的大小.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "椭圆" ], "genre": "解答题", "ans": "", @@ -256553,7 +257382,8 @@ "content": "水星的运行轨道是以太阳的中心为一个焦点的椭圆, 轨道上离太阳中心最近的距离约为$4.7\\times 10^8\\text{km}$, 最远的距离约为$7.05\\times 10^8\\text{km}$. 以这个轨道的中心为原点, 以太阳中心及轨道中心所在直线为$x$轴, 建立平面直角坐标系. 求水星运行轨道的方程. (长半轴的长和短半轴的长精确到$0. 1\\times 10^8\\text{km}$)", "objs": [], "tags": [ - "第七单元" + "第七单元", + "椭圆" ], "genre": "解答题", "ans": "", @@ -256574,7 +257404,8 @@ "content": "双曲线$\\dfrac{x^2}{64}-\\dfrac{y^2}{36}=1$上一点$P$到焦点$F_1$的距离等于$6$, 求点$P$到另一焦点$F_2$的距离.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "双曲线" ], "genre": "解答题", "ans": "", @@ -256595,7 +257426,8 @@ "content": "已知双曲线以坐标轴为对称轴, 两个顶点间的距离为$2$, 焦点到渐近线的距离为$\\sqrt 2$. 求该双曲线的方程.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "双曲线" ], "genre": "解答题", "ans": "", @@ -256616,7 +257448,8 @@ "content": "如果双曲线关于原点对称, 它的焦点在坐标轴上, 实轴的长为$8$, 焦距为$10$. 写出此双曲线的方程.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "双曲线" ], "genre": "解答题", "ans": "", @@ -256637,7 +257470,8 @@ "content": "如果方程$\\dfrac{x^2}{m+2}-\\dfrac{y^2}{m+1}=1$表示焦点在$y$轴上的双曲线, 求实数$m$的取值范围.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "双曲线" ], "genre": "解答题", "ans": "", @@ -256660,7 +257494,8 @@ "content": "已知双曲线经过点$(1, 1)$, 其渐近线方程为$y=\\pm\\sqrt 2x$. 求此双曲线的方程.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "双曲线" ], "genre": "解答题", "ans": "", @@ -256681,7 +257516,8 @@ "content": "已知双曲线的中心在原点, 焦点在$y$轴上, 并且双曲线上两点$P_1$、$P_2$的坐标分别为$(3, -4\\sqrt 2)$、$(\\dfrac 94, 5)$. 求该双曲线的方程.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "双曲线" ], "genre": "解答题", "ans": "", @@ -256702,7 +257538,8 @@ "content": "已知离心率为$\\dfrac 53$的双曲线与椭圆$\\dfrac{x^2}{40}+\\dfrac{y^2}{15}=1$有公共焦点, 求此双曲线的方程.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "双曲线" ], "genre": "解答题", "ans": "", @@ -256744,7 +257581,8 @@ "content": "求抛物线$y^2=ax$($a\\ne 0$)的焦点坐标和准线方程.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "抛物线" ], "genre": "解答题", "ans": "", @@ -256767,7 +257605,8 @@ "content": "若抛物线$y^2=2x$上的$A$、$B$两点到焦点$F$的距离之和是$5$, 求线段$AB$的中点的横坐标.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "抛物线" ], "genre": "解答题", "ans": "", @@ -256790,7 +257629,8 @@ "content": "求以坐标原点为顶点, 以$y$轴为对称轴, 并经过点$P(-6, -3)$的抛物线的标准方程.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "抛物线" ], "genre": "解答题", "ans": "", @@ -256811,7 +257651,8 @@ "content": "已知直线$y=kx-4$与抛物线$y^2=8x$有且只有一个公共点, 求实数$k$的值.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "抛物线" ], "genre": "解答题", "ans": "", @@ -256836,7 +257677,8 @@ "content": "已知一隧道的顶部是抛物拱形, 拱高是$5\\text{m}$, 跨度为$10\\text{m}$. 建立适当的平面直角坐标系, 求此拱形所在的抛物线方程.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "抛物线" ], "genre": "解答题", "ans": "", @@ -256878,7 +257720,8 @@ "content": "过抛物线$y^2=2px$($p>0$)焦点的一条直线与抛物线相交于两个不同的点, 求证: 这两个点的纵坐标$y_1$、$y_2$满足$y_1y_2=-p^2$.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "抛物线" ], "genre": "解答题", "ans": "", @@ -256899,7 +257742,8 @@ "content": "过抛物线$y^2=2px$的焦点且倾斜角为$\\alpha$的直线$l$与抛物线交于$A$、$B$两点, 求证:$|AB|=\\dfrac{2p}{\\sin^2\\alpha}$.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "抛物线" ], "genre": "解答题", "ans": "", @@ -256920,7 +257764,8 @@ "content": "写出椭圆方程推导过程中的``反过来推演'', 即验证:若点$M$以方程$\\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1$($a>b>0$)的解$(x, y)$为坐标, 则点$M$一定在以$F_1(-c, 0)$与$F_2(c, 0)$为焦点的椭圆上, 这里$c=\\sqrt{a^2-b^2}$.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "椭圆" ], "genre": "解答题", "ans": "", @@ -257076,7 +257921,8 @@ "content": "求过点$M(2, \\dfrac \\pi 2)$且平行于极轴的直线的极坐标方程.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "直线" ], "genre": "解答题", "ans": "", @@ -257165,7 +258011,8 @@ "content": "证明:椭圆$C_1: \\dfrac{x^2}4+\\dfrac{y^2}3=1$与椭圆$C_2: \\dfrac{x^2}3+\\dfrac{y^2}4=1$的四个交点共圆.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "椭圆" ], "genre": "解答题", "ans": "", @@ -257186,7 +258033,8 @@ "content": "点$P$在椭圆$\\dfrac{x^2}4+y^2=1$上运动, 求它到直线$l: x+2y-2=0$的距离的最大值.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "椭圆" ], "genre": "解答题", "ans": "", @@ -257207,7 +258055,10 @@ "content": "点$P$到定点$F(2, 0)$的距离与它到直线$x=8$的距离之比为$k$, 请分别给出$k$的某个值, 使得轨迹是椭圆、双曲线和抛物线.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "椭圆", + "双曲线", + "抛物线" ], "genre": "解答题", "ans": "", @@ -257228,7 +258079,8 @@ "content": "已知椭圆$C: \\dfrac{x^2}4+\\dfrac{y^2}3=1$, 试确定$m$的取值范围, 使该椭圆上有两个不同的点关于直线$l: y=4x+m$对称.", "objs": [], "tags": [ - "第七单元" + "第七单元", + "椭圆" ], "genre": "解答题", "ans": "", @@ -287921,6 +288773,63 @@ "remark": "", "space": "12ex" }, + "012030": { + "id": "012030", + "content": "点$A(-4,2)$是抛物线$y^2=-8x$内一点, 抛物线上的点$M$到$A$点的距离与它到焦点的距离之和最小, 则点$M$的坐标是\\blank{50}, 最小距离是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第一轮复习讲义", + "edit": [ + "20221105\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012031": { + "id": "012031", + "content": "设$F$为抛物线$y^2=4x$的焦点, $ABC$为该抛物线上三点.若$\\overrightarrow{FA}+\\overrightarrow{FB}+\\overrightarrow{FC}=\\overrightarrow 0$, 则$|\\overrightarrow{FA}|+|\\overrightarrow{FB}|+|\\overrightarrow{FC}|=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第一轮复习讲义", + "edit": [ + "20221105\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012032": { + "id": "012032", + "content": "设抛物线$y^2=2x$的焦点为$F$, 过点$M(\\sqrt 3,0)$的直线与抛物线相交于$AB$两点, 与抛物线的准线相交于$C$, $| BF |=2$, 则$\\triangle BCF$与$\\triangle ACF$的面积之比为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第一轮复习讲义", + "edit": [ + "20221105\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, "020001": { "id": "020001", "content": "判断下列各组对象能否组成集合, 若能组成集合, 指出是有限集还是无限集.\\\\\n(1) 上海市控江中学$2022$年入学的全体高一年级新生;\\\\\n(2) 中国现有各省的名称;\\\\\n(3) 太阳、$2$、上海市;\\\\\n(4) 大于$10$且小于$15$的有理数;\\\\\n(5) 末位是$3$的自然数;\\\\\n(6) 影响力比较大的中国数学家;\\\\\n(7) 方程$x^2+x-3=0$的所有实数解;\\\\ \n(8) 函数$y=\\dfrac 1x$图像上所有的点;\\\\ \n(9) 在平面直角坐标系中, 到定点$(0, 0)$的距离等于$1$的所有点;\\\\\n(10) 不等式$3x-10<0$的所有正整数解;\\\\\n(11) 所有的平面四边形.", @@ -294022,7 +294931,8 @@ "content": "已知直线$l$经过点$(-\\sqrt{5},0)$且法向量为$(1,2)$, 则原点$O$到直线$l$的距离为\\blank{50}.", "objs": [], "tags": [ - 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"genre": "", + "genre": "选择题", "ans": "", "solution": "", "duration": -1, @@ -305975,7 +306885,7 @@ "第六单元", "空间向量" ], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -305987,7 +306897,7 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "030470": { "id": "030470", @@ -305999,7 +306909,7 @@ "第六单元", "空间向量" ], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -306023,7 +306933,7 @@ "第六单元", "空间向量" ], - "genre": "", + "genre": "选择题", "ans": "", "solution": "", "duration": -1, @@ -306049,7 +306959,7 @@ "第六单元", "空间向量" ], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -306061,7 +306971,7 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "030473": { "id": "030473",