From 08ad9a29455ceccf87d3d64867bbd9aa01ee9fc7 Mon Sep 17 00:00:00 2001 From: "weiye.wang" Date: Sat, 24 Feb 2024 19:23:48 +0800 Subject: [PATCH] =?UTF-8?q?=E5=BD=95=E5=85=A52025=E5=B1=8A=E9=AB=98?= =?UTF-8?q?=E4=BA=8C=E4=B8=8B=E5=AD=A6=E6=9C=9F=E6=A0=A1=E6=9C=AC=E4=BD=9C?= =?UTF-8?q?=E4=B8=9A-=E8=A7=A3=E6=9E=90=E5=87=A0=E4=BD=95=E6=96=B0?= =?UTF-8?q?=E9=A2=98?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- 工具v2/文本文件/新题收录列表.txt | 3 + 题库0.3/Problems.json | 1575 +++++++++++++++++++++++++++++- 2 files changed, 1544 insertions(+), 34 deletions(-) diff --git a/工具v2/文本文件/新题收录列表.txt b/工具v2/文本文件/新题收录列表.txt index 7c4c3447..f3e2837a 100644 --- a/工具v2/文本文件/新题收录列表.txt +++ b/工具v2/文本文件/新题收录列表.txt @@ -364,3 +364,6 @@ 20240224-134951 2024年空中课堂必修第一册补充例题与习题 024597:024598,014651,024599:024602,000007:000008,000025,000027,000029,000009,000014:000016,000031,000037,000039:000040,024603:024604,000052:000053,000056:000058,000062:000065,000067,000069:000073,024605:024609,000048,000054:000055,000061,000066,000081:000084,000087:000088,007939,000090,024610,030726,024611:024612,007924,000077:000080,000085:000086 +20240224-192316 2025届高二下学期校本作业-解析几何 +014469,040971,002106,040972:040975,008878,008877,014470,040976:040979,008882,021195:021196,040980:040982,008892,040983:040986,021183,008883,040987:040989,008895,040990:040992,008898,040993,008891,021197,021184:021187,021198,021188,040994,021190:021191,040995:040996,021204:021209,040997:040998,021212,021200:021203,040999:041000,021222:021233,041001:041003,021242:021244,041004:041005,021251,021263,021252:021254,008917,021255:021256,041006,021258:021261,021267:021268,021270:021272,041007,021276,021279,021284,021269,021275,041008:041011,008929,041012,021278,041013:041014,021280,041015,021304,021308,021287,009840,021309,021290:021291,041016,021339,021289,021293:021295,021305,013106,021292,008930,008934,008922,021299:021300,021321,041017:041018,021316,021326,008925,021319,010689,041019:041021,021331,041022:041023,021334:021338,021340:021343,041024,008846:008847,008852:008853,041025:041027,010704,010703,021348:021349,021351,041028,021352:021353,041029,021354:021355,009845:009846,021358:021359,021362:021364,012470,041030:041039 + diff --git a/题库0.3/Problems.json b/题库0.3/Problems.json index 5581e0b5..b38f8ef7 100644 --- a/题库0.3/Problems.json +++ b/题库0.3/Problems.json @@ -75145,7 +75145,8 @@ ], "same": [], "related": [ - "002307" + "002307", + "040971" ], "remark": "", "space": "", @@ -76161,7 +76162,8 @@ ], "same": [], "related": [ - "040968" + "040968", + "041034" ], "remark": "", "space": "", @@ -77721,7 +77723,8 @@ ], "same": [], "related": [ - "024124" + "024124", + "041013" ], "remark": "", "space": "", @@ -107192,7 +107195,9 @@ "20220701\t王伟叶" ], "same": [], - "related": [], + "related": [ + "040995" + ], "remark": "", "space": "4em", "unrelated": [] @@ -251687,7 +251692,9 @@ "20220726\t王伟叶" ], "same": [], - "related": [], + "related": [ + "040977" + ], "remark": "", "space": "4em", "unrelated": [] @@ -251709,7 +251716,9 @@ "20220726\t王伟叶" ], "same": [], - "related": [], + "related": [ + "040980" + ], "remark": "", "space": "4em", "unrelated": [] @@ -251759,7 +251768,9 @@ "20220726\t王伟叶" ], "same": [], - "related": [], + "related": [ + "040983" + ], "remark": "", "space": "4em", "unrelated": [] @@ -251836,7 +251847,9 @@ "20220726\t王伟叶" ], "same": [], - "related": [], + "related": [ + "040981" + ], "remark": "", "space": "4em", "unrelated": [] @@ -251888,7 +251901,9 @@ "20220726\t王伟叶" ], "same": [], - "related": [], + "related": [ + "040979" + ], "remark": "", "space": "", "unrelated": [] @@ -252062,7 +252077,9 @@ "20220726\t王伟叶" ], "same": [], - "related": [], + "related": [ + "040992" + ], "remark": "", "space": "4em", "unrelated": [] @@ -252193,7 +252210,9 @@ "20220726\t王伟叶" ], "same": [], - "related": [], + "related": [ + "040985" + ], "remark": "", "space": "", "unrelated": [] @@ -252291,7 +252310,9 @@ "20220726\t王伟叶" ], "same": [], - "related": [], + "related": [ + "040990" + ], "remark": "", "space": "", "unrelated": [] @@ -252903,7 +252924,8 @@ "021240" ], "related": [ - "023707" + "023707", + "041003" ], "remark": "", "space": "4em", @@ -253136,7 +253158,8 @@ ], "same": [], "related": [ - "010682" + "010682", + "041008" ], "remark": "", "space": "4em", @@ -299983,7 +300006,8 @@ ], "same": [], "related": [ - "008924" + "008924", + "041008" ], "remark": "", "space": "4em", @@ -300032,7 +300056,9 @@ "same": [ "008923" ], - "related": [], + "related": [ + "041010" + ], "remark": "", "space": "4em", "unrelated": [] @@ -459623,7 +459649,9 @@ "20230503\t王伟叶" ], "same": [], - "related": [], + "related": [ + "040989" + ], "remark": "", "space": "", "unrelated": [] @@ -510900,7 +510928,9 @@ "20230707\t王伟叶" ], "same": [], - "related": [], + "related": [ + "041014" + ], "remark": "", "space": "4em", "unrelated": [] @@ -527739,7 +527769,9 @@ "20230730\t王伟叶" ], "same": [], - "related": [], + "related": [ + "041027" + ], "remark": "", "space": "", "unrelated": [] @@ -575207,7 +575239,8 @@ ], "same": [], "related": [ - "023697" + "023697", + "040994" ], "remark": "", "space": "4em", @@ -575432,7 +575465,9 @@ "20230101\t王伟叶" ], "same": [], - "related": [], + "related": [ + "040996" + ], "remark": "", "space": "4em", "unrelated": [] @@ -575679,7 +575714,9 @@ "20230101\t王伟叶" ], "same": [], - "related": [], + "related": [ + "040997" + ], "remark": "", "space": "", "unrelated": [] @@ -575701,7 +575738,9 @@ "20230101\t王伟叶" ], "same": [], - "related": [], + "related": [ + "040998" + ], "remark": "", "space": "4em", "unrelated": [] @@ -576222,7 +576261,9 @@ "20230101\t王伟叶" ], "same": [], - "related": [], + "related": [ + "041001" + ], "remark": "", "space": "", "unrelated": [] @@ -576553,7 +576594,9 @@ "20230101\t王伟叶" ], "same": [], - "related": [], + "related": [ + "041005" + ], "remark": "", "space": "", "unrelated": [] @@ -576751,7 +576794,9 @@ "20230101\t王伟叶" ], "same": [], - "related": [], + "related": [ + "041006" + ], "remark": "", "space": "", "unrelated": [] @@ -577403,7 +577448,9 @@ "20230101\t王伟叶" ], "same": [], - "related": [], + "related": [ + "041015" + ], "remark": "", "space": "", "unrelated": [] @@ -577853,7 +577900,9 @@ "20230101\t王伟叶" ], "same": [], - "related": [], + "related": [ + "041016" + ], "remark": "", "space": "", "unrelated": [] @@ -577875,7 +577924,9 @@ "20230101\t王伟叶" ], "same": [], - "related": [], + "related": [ + "041018" + ], "remark": "", "space": "", "unrelated": [] @@ -577965,7 +578016,9 @@ "20230101\t王伟叶" ], "same": [], - "related": [], + "related": [ + "041019" + ], "remark": "", "space": "4em", "unrelated": [] @@ -578488,7 +578541,9 @@ "20230101\t王伟叶" ], "same": [], - "related": [], + "related": [ + "041023" + ], "remark": "", "space": "", "unrelated": [] @@ -578841,7 +578896,9 @@ "20230101\t王伟叶" ], "same": [], - "related": [], + "related": [ + "041028" + ], "remark": "", "space": "", "unrelated": [] @@ -657088,7 +657145,8 @@ ], "same": [], "related": [ - "021201" + "021201", + "041032" ], "remark": "", "space": "", @@ -748767,7 +748825,9 @@ "20240221\t杨懿荔" ], "same": [], - "related": [], + "related": [ + "040982" + ], "remark": "", "space": "", "unrelated": [] @@ -749257,5 +749317,1452 @@ "remark": "", "space": "4em", "unrelated": [] + }, + "040971": { + "id": "040971", + "content": "椭圆 $\\dfrac{x^2}{16}+\\dfrac{y^2}{9}=1$ 的焦距是\\blank{50}, 焦点坐标是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高二下学期校本作业-解析几何", + "edit": [ + "20240224\t杨懿荔" + ], + "same": [], + "related": [ + "002308" + ], + "remark": "", + "space": "", + "unrelated": [] + }, + "040972": { + "id": "040972", + "content": "已知 $A(0,3)$、$B(0,-3)$ 两点. 若动点 $P$ 满足 $|PA|+|PB|=6$, 则点 $P$ 的轨迹方程是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高二下学期校本作业-解析几何", + "edit": [ + "20240224\t杨懿荔" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "040973": { + "id": "040973", + "content": "已知 $A(0,3)$、$B(0,-3)$ 两点. 若动点 $P$ 满足 $|PA|+|PB|=8$, 则点 $P$ 的轨迹为\\bracket{20}.\n\\fourch{焦点在 $x$ 轴上的椭圆}{焦点在 $y$ 轴上的椭圆}{$x$ 轴上的线段}{$y$ 轴上的线段}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高二下学期校本作业-解析几何", + "edit": [ + "20240224\t杨懿荔" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "040974": { + "id": "040974", + "content": "已知 $A(0,3)$、$B(0,-3)$ 两点. 若动点 $P$ 满足 $|PA|+|PB|=8$, 则点 $P$ 的轨迹方程是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高二下学期校本作业-解析几何", + "edit": [ + "20240224\t杨懿荔" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "040975": { + "id": "040975", + "content": "``$m>2$''是``方程 $\\dfrac{x^2}{m-2}+\\dfrac{y^2}{5-m}=1$ 表示的曲线是椭圆''的\\bracket{20}.\n\\fourch{充分非必要条件}{必要非充分条件}{充要条件}{既非充分又非必要条件}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高二下学期校本作业-解析几何", + "edit": [ + "20240224\t杨懿荔" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "040976": { + "id": "040976", + "content": "椭圆的中心在原点, 焦距为 6 , 且经过点 $(0,4)$, 则它的标准方程是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高二下学期校本作业-解析几何", + "edit": [ + "20240224\t杨懿荔" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "040977": { + "id": "040977", + "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.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高二下学期校本作业-解析几何", + "edit": [ + "20240224\t杨懿荔" + ], + "same": [], + "related": [ + "008873" + ], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "040978": { + "id": "040978", + "content": "(1) 已知方程 $\\dfrac{x^2}{k-4}+\\dfrac{y^2}{10-k}=1$ 表示圆, 求实数 $k$ 的值;\\\\\n(2) 已知方程 $\\dfrac{x^2}{k-4}+\\dfrac{y^2}{10-k}=1$ 表示焦点在 $y$ 轴上的椭圆, 求实数 $k$ 的取值范围.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高二下学期校本作业-解析几何", + "edit": [ + "20240224\t杨懿荔" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "040979": { + "id": "040979", + "content": "若椭圆的方程为 $16 x^2+25 y^2=400$, 则此椭圆的长半轴长为\\blank{50}, 短轴长为\\blank{50}, 焦距为\\blank{50}, 离心率为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高二下学期校本作业-解析几何", + "edit": [ + "20240224\t杨懿荔" + ], + "same": [], + "related": [ + "008881" + ], + "remark": "", + "space": "", + "unrelated": [] + }, + "040980": { + "id": "040980", + "content": "已知动点 $M$ 到定点 $A(-\\dfrac{9}{4}, 0)$ 与 $B(\\dfrac{9}{4}, 0)$ 的距离的和是 $\\dfrac{25}{2}$, 则点 $M$ 的轨迹方程是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高二下学期校本作业-解析几何", + "edit": [ + "20240224\t杨懿荔" + ], + "same": [], + "related": [ + "008874" + ], + "remark": "", + "space": "", + "unrelated": [] + }, + "040981": { + "id": "040981", + "content": "已知椭圆 $C$ 的两个焦点分别为 $F_1(-3,0)$、$F_2(3,0)$, 若其离心率为\\blank{50}, 则椭圆 $C$ 的方程为 $\\dfrac{x^2}{25}+\\dfrac{y^2}{16}=1$.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高二下学期校本作业-解析几何", + "edit": [ + "20240224\t杨懿荔" + ], + "same": [], + "related": [ + "008879" + ], + "remark": "", + "space": "", + "unrelated": [] + }, + "040982": { + "id": "040982", + "content": "焦距为 6 的椭圆上一点到其两个焦点的距离之和为 10 , 则此椭圆的标准方程为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高二下学期校本作业-解析几何", + "edit": [ + "20240224\t杨懿荔" + ], + "same": [], + "related": [ + "040949" + ], + "remark": "", + "space": "", + "unrelated": [] + }, + "040983": { + "id": "040983", + "content": "若椭圆 $\\dfrac{x^2}{25}+\\dfrac{y^2}{9}=1$ 的两个焦点分别为 $F_1$、$F_2$, 点 $P$ 为此椭圆上的任意一点, 则 $\\triangle PF_1F_2$的周长为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高二下学期校本作业-解析几何", + "edit": [ + "20240224\t杨懿荔" + ], + "same": [], + "related": [ + "008876" + ], + "remark": "", + "space": "", + "unrelated": [] + }, + "040984": { + "id": "040984", + "content": "如果椭圆 $5 x^2+k y^2=5$ 的一个焦点是 $(0,2)$, 那么 $k=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高二下学期校本作业-解析几何", + "edit": [ + "20240224\t杨懿荔" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "040985": { + "id": "040985", + "content": "已知点 $P$ 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,\\\\\n(1) 求 $|PF_1|$ 的最大值;\\\\\n(2) 求 $|PF_1|$ 的最小值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高二下学期校本作业-解析几何", + "edit": [ + "20240224\t杨懿荔" + ], + "same": [], + "related": [ + "008893" + ], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "040986": { + "id": "040986", + "content": "写出分别满足下列条件的椭圆的标准方程:\\\\\n(1) 焦距等于 4 , 且经过点 $P(\\dfrac{2 \\sqrt{6}}{3},-\\dfrac{2 \\sqrt{6}}{3})$.\\\\\n(2) 过 $(-\\dfrac{3}{2}, \\dfrac{5}{2})$ 与 $(\\sqrt{3}, \\sqrt{5})$ 两点.\\\\\n(3) 长轴长是短轴长的 2 倍, 且过点 $(-2,-4)$.\\\\\n(4) 椭圆的一个顶点和一个焦点分别是直线 $x+3 y-6=0$ 与两坐标轴的交点.\\\\\n(5) 已知椭圆在 $x$ 轴上的一个焦点 $F$ 与短轴 $B_1B_2$ 两端点的连线互相垂直, 且点 $F$ 和长轴上较近的端点 $A$ 的距离是 $\\sqrt{10}-\\sqrt{5}$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高二下学期校本作业-解析几何", + "edit": [ + "20240224\t杨懿荔" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "040987": { + "id": "040987", + "content": "在平面直角坐标系中, 椭圆 $C$ 的中心为原点, 焦点 $F_1$、$F_2$ 在 $x$ 轴上, 且 $\\dfrac{c}{a}=\\dfrac{\\sqrt{2}}{2}$, 如果经过 $F_1$ 的直线 $l$ 交椭圆 $C$ 于 $A$、$B$ 两点, 且 $\\triangle ABF_2$ 的周长为 16 , 那么椭圆 $C$ 的方程为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高二下学期校本作业-解析几何", + "edit": [ + "20240224\t杨懿荔" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "040988": { + "id": "040988", + "content": "已知两点 $B(-3,0)$、$C(3,0)$. 若 $\\triangle ABC$ 的周长为 20 , 则顶点 $A$ 的轨迹方程为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高二下学期校本作业-解析几何", + "edit": [ + "20240224\t杨懿荔" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "040989": { + "id": "040989", + "content": "已知椭圆 $\\dfrac{x^2}{9}+\\dfrac{y^2}{2}=1$ 的焦点为 $F_1$、$F_2$, 点 $P$ 在椭圆上, 若 $|PF_1|=4$, 则 $|PF_2|=\\blank{50}, \\angle F_1PF_2$ 的大小为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高二下学期校本作业-解析几何", + "edit": [ + "20240224\t杨懿荔" + ], + "same": [], + "related": [ + "016667" + ], + "remark": "", + "space": "", + "unrelated": [] + }, + "040990": { + "id": "040990", + "content": "在 $\\triangle ABC$ 中, 已知 $A(-1,0)$、$C(1,0)$. 若 $a>b>c$, 且满足 $2 \\sin B=\\sin A+\\sin C$, 则顶点 $B$ 的轨迹的方程是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高二下学期校本作业-解析几何", + "edit": [ + "20240224\t杨懿荔" + ], + "same": [], + "related": [ + "008896" + ], + "remark": "", + "space": "", + "unrelated": [] + }, + "040991": { + "id": "040991", + "content": "已知椭圆 $\\dfrac{x^2}{12}+\\dfrac{y^2}{3}=1$ 的焦点为 $F_1$、$F_2$, 点 $P$ 在此椭圆上.若线段 $PF_1$ 的中点 $M$ 恰在 $y$轴上, 则 $|PF_1|$ 是 $|PF_2|$ 的\\bracket{20}.\n\\fourch{7 倍}{5 倍}{4 倍}{3 倍}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高二下学期校本作业-解析几何", + "edit": [ + "20240224\t杨懿荔" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "040992": { + "id": "040992", + "content": "$\\triangle ABC$ 的两个顶点 $A$、$B$ 的坐标分别是 $(-6,0)$、$(6,0), AC$、$BC$ 边所在直线的斜率之积等于 $-\\dfrac{4}{9}$, 求顶点 $C$ 的轨迹方程.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高二下学期校本作业-解析几何", + "edit": [ + "20240224\t杨懿荔" + ], + "same": [], + "related": [ + "008888" + ], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "040993": { + "id": "040993", + "content": "已知 $P$ 是椭圆 $\\dfrac{x^2}{25}+\\dfrac{y^2}{16}=1$ 上的一点, $F_1, F_2$ 是椭圆的两个焦点, 且 $\\angle F_1PF_2=30^{\\circ}$,求 $\\Delta F_1PF_2$ 的面积.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高二下学期校本作业-解析几何", + "edit": [ + "20240224\t杨懿荔" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "040994": { + "id": "040994", + "content": "椭圆 $\\dfrac{x^2}{25}+\\dfrac{y^2}{9}=1$ 上一点 $M$ 到左焦点 $F_1$ 的距离为 $2, N$ 是 $MF_1$ 的中点, $O$ 是坐标原点,则 $|ON|$ 的长为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高二下学期校本作业-解析几何", + "edit": [ + "20240224\t杨懿荔" + ], + "same": [], + "related": [ + "021189" + ], + "remark": "", + "space": "", + "unrelated": [] + }, + "040995": { + "id": "040995", + "content": "已知椭圆 $\\dfrac{x^2}{2}+y^2=1$.\\\\\n(1) 求过点 $M(\\dfrac{1}{2}, \\dfrac{1}{2})$ 且被 $M$ 平分的弦所在直线方程;\\\\\n(2) 求斜率为 2 的平行弦中点 $P$ 的轨迹方程;\\\\\n(3) 过椭圆的左焦点 $F_1$ 引椭圆的割线, 求截得的弦的中点 $Q$ 的轨迹方程.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高二下学期校本作业-解析几何", + "edit": [ + "20240224\t杨懿荔" + ], + "same": [], + "related": [ + "003408" + ], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "040996": { + "id": "040996", + "content": "直线 $y=k x-2$ 交椭圆 $\\dfrac{x^2}{80}+\\dfrac{y^2}{20}=1$ 于不同两点 $P$ 和 $Q$, 若弦 $PQ$ 的中点的横坐标等于 2 ,则弦长 $|PQ|=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高二下学期校本作业-解析几何", + "edit": [ + "20240224\t杨懿荔" + ], + "same": [], + "related": [ + "021199" + ], + "remark": "", + "space": "", + "unrelated": [] + }, + "040997": { + "id": "040997", + "content": "若 $AB$ 为过椭圆 $\\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1$($a>b>0$) 中心的弦, $F_1$ 为此椭圆的焦点, 则 $\\Delta F_1AB$ 的面积的最大值为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高二下学期校本作业-解析几何", + "edit": [ + "20240224\t杨懿荔" + ], + "same": [], + "related": [ + "021210" + ], + "remark": "", + "space": "", + "unrelated": [] + }, + "040998": { + "id": "040998", + "content": "已知 $F_1$、$F_2$ 分别为椭圆 $\\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1$($a>b>0$) 的左、右焦点, 点 $P$ 在椭圆上, $O$ 为坐标原点, $\\triangle POF_2$ 是面积为 $4 \\sqrt{3}$ 的正三角形, 则 $b^2$ 的值为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高二下学期校本作业-解析几何", + "edit": [ + "20240224\t杨懿荔" + ], + "same": [], + "related": [ + "021211" + ], + "remark": "", + "space": "", + "unrelated": [] + }, + "040999": { + "id": "040999", + "content": "填空题:\\\\\n(1) 若 $P$ 是双曲线 $x^2-y^2=9$ 的左支上一点, $F_1$、$F_2$ 分别是双曲线的左、右焦点, 则 $|PF_1|-|PF_2|=$\\blank{50}.\\\\\n(2) 设 $a$ 是正实数. 已知点 $P$ 与 $A(-a, 0)$、$B(a, 0)$ 两定点的连线的斜率之积为定值 $t$($t>0$), 则点 $P$ 的轨迹方程是\\blank{50}.\\\\\n(3) 若椭圆 $\\dfrac{x^2}{4}+\\dfrac{y^2}{2 k^2}=1$ 与双曲线 $\\dfrac{x^2}{k^2}-\\dfrac{y^2}{2}=1$ 的焦点相同, 则正数 $k$ 的值是\\blank{50}.\\\\\n(4) 若椭圆 $\\dfrac{x^2}{4}+\\dfrac{y^2}{2 k^2}=1$ 与双曲线 $\\dfrac{x^2}{k^2}-\\dfrac{y^2}{2}=1$ 的焦距相同, 则正数 $k$ 的值是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高二下学期校本作业-解析几何", + "edit": [ + "20240224\t杨懿荔" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "041000": { + "id": "041000", + "content": "选择题:\\\\\n(1) 已知 $F_1(0,-5)$、$F_2(0,5)$ 两点, 若动点 $P$ 满足 $|PF_1|-|PF_2|=8$, 则点 $P$ 的轨迹是\\bracket{20}.\n\\fourch{双曲线}{双曲线靠近 $F_1$ 的一支}{双曲线靠近 $F_2$ 的一支}{一条射线}\\\\\n(2) 在 $\\triangle ABC$ 中, 已知 $A(-4,0)$、$B(4,0)$ 两点. 若 $\\sin A-\\sin B=\\dfrac{1}{2}\\sin C$, 则顶点 $C$ 的轨迹方程是\\bracket{20}.\n\\fourch{$\\dfrac{x^2}{4}-\\dfrac{y^2}{12}=1$($x<-2$)}{$\\dfrac{x^2}{4}-\\dfrac{y^2}{12}=1$($x>2$)}{$\\dfrac{x^2}{12}-\\dfrac{y^2}{4}=1$($x>2 \\sqrt{3}$)}{$\\dfrac{x^2}{12}-\\dfrac{y^2}{4}=1$($y \\neq 0$)}\\\\\n(3) 已知椭圆 $\\dfrac{x^2}{m^2}+\\dfrac{y^2}{n^2}=1$($|m|>|n|$) 和双曲线 $\\dfrac{x^2}{a^2}-\\dfrac{y^2}{b^2}=1$ 相同的两焦点 $F_1$、$F_2$, 若点 $P$ 为两曲线的一个交点, 则 $|PF_1| \\cdot|PF_2|$ 等于\\bracket{20}.\n\\fourch{$m^2-a^2$}{$\\dfrac{1}{2}(m^2-a^2)$}{$m-n$}{$a^2-m^2$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高二下学期校本作业-解析几何", + "edit": [ + "20240224\t杨懿荔" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "041001": { + "id": "041001", + "content": "填空题:\\\\\n(1) 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}, 离心率为\\blank{50}.\\\\\n(2) 双曲线 $2 x^2-y^2=8$ 的焦点坐标是\\blank{50}, 两条渐近线的夹角为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高二下学期校本作业-解析几何", + "edit": [ + "20240224\t杨懿荔" + ], + "same": [], + "related": [ + "021234" + ], + "remark": "", + "space": "", + "unrelated": [] + }, + "041002": { + "id": "041002", + "content": "选择题:\\\\\n(1) 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{$\\dfrac{x^2}{9}-\\dfrac{y^2}{16}=1$}{$\\dfrac{x^2}{36}-\\dfrac{y^2}{11}=1$}{$\\dfrac{x^2}{16}-\\dfrac{y^2}{9}=1$ }{$\\dfrac{x^2}{11}-\\dfrac{y^2}{36}=1$}\\\\\n(2) 在下列双曲线中, 以 $y= \\pm \\dfrac{1}{2}x$ 为渐近线的是\\bracket{20}.\n\\fourch{$\\dfrac{x^2}{16}-\\dfrac{y^2}{4}=1$ ,}{$\\dfrac{x^2}{4}-\\dfrac{y^2}{16}=1$ }{$\\dfrac{x^2}{2}-y^2=1$ ,}{$x^2-\\dfrac{y^2}{2}=1$}\\\\\n(3) 若方程 $4 x^2+k y^2=4 k$ 表示双曲线, 则此双曲线的虚轴长等于\n\\fourch{$2 \\sqrt{k}$}{$2 \\sqrt{-k}$}{$\\sqrt{k}$}{$\\sqrt{-k}$}\\\\\n(4) 已知双曲线 $\\dfrac{x^2}{a^2}-\\dfrac{y^2}{b^2}=1$($a>0$, $b>0$) 的两焦点为 $F_1$、$F_2$, 若弦 $AB$ 经过点 $F_1$, 且\n$A$、$B$ 均在此双曲线的左支上, $|AB|=l$, 则 $\\triangle ABF_2$ 的周长为\\bracket{20}.\n\\fourch{$4 a+l$ }{$4 a+2 l$}{$2 a+l$ }{$4 a-l$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高二下学期校本作业-解析几何", + "edit": [ + "20240224\t杨懿荔" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "041003": { + "id": "041003", + "content": "(1)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.\\\\\n(2) 已知双曲线的虚轴的长为 6 , 一条渐近线的方程为 $3 x-y=0$, 求此双曲线的标准方程.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高二下学期校本作业-解析几何", + "edit": [ + "20240224\t杨懿荔" + ], + "same": [], + "related": [ + "008916" + ], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "041004": { + "id": "041004", + "content": "填空题:\\\\\n(1) 已知双曲线 $C$ 的中心在坐标原点, 若它的一个焦点为 $(3,0)$, 一条渐近线的方程为 $2 x-3 y=0$, 则此双曲线的方程为\\blank{50}.\\\\\n(2) 若双曲线的两渐近线的夹角为 $\\dfrac{\\pi}{3}$, 则焦距与实轴长之比 $\\dfrac{c}{a}=$\\blank{50}.\\\\\n(3) 已知 $F_1(-3,0)$、$F_2(3,0)$ 两点, 满足条件 $|MF_1|+|MF_2|=2 m+1$ 的动点 $M$ 的轨迹是椭圆, 满足条件 $|NF_1|-|NF_2|=2 m-1$ 的动点 $N$ 的轨迹是双曲线, 则实数 $m$ 的取值范围是\\blank{50}.\\\\\n(4) 若双曲线 $\\dfrac{x^2}{4}-\\dfrac{y^2}{m}=1$ 的渐近线方程为 $y= \\pm \\dfrac{\\sqrt{3}}{2}x$, 则此双曲线的焦点坐标是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高二下学期校本作业-解析几何", + "edit": [ + "20240224\t杨懿荔" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "041005": { + "id": "041005", + "content": "选择题:\\\\\n(1) 若 $a b \\neq 0$ 且 $a \\neq b$, 则 $a x-y+b=0$ 和 $b x^2+a y^2=a b$ 所表示的曲线只可能是下图中的\\bracket{20}.\n\\fourch{\\begin{tikzpicture}[>=latex,scale = 0.4]\n\\draw [->] (-3,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,-3) -- (0,3) node [left] {$y$};\n\\draw (0,0) node [below right] {$O$};\n\\draw (0,0) ellipse (1 and 2);\n\\draw (-2.5,3) -- (1.5,-3);\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex,scale = 0.4]\n\\draw [->] (-3,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,-3) -- (0,3) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = -3:3] plot (\\x,{sqrt(1+\\x*\\x/2)});\n\\draw [domain = -3:3] plot (\\x,{-sqrt(1+\\x*\\x/2)});\n\\draw (-3,-0.5) -- (0.5,3);\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex,scale = 0.4]\n\\draw [->] (-3,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,-3) -- (0,3) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (0,0) ellipse (2 and 1);\n\\draw (-1.5,3) -- (2.5,-3);\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex,scale = 0.4]\n\\draw [->] (-3,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,-3) -- (0,3) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = -3:3] plot ({sqrt(0.5+0.25*\\x*\\x)},\\x);\n\\draw [domain = -3:3] plot ({-sqrt(0.5+0.25*\\x*\\x)},\\x);\n\\draw (-2,-3) -- (3,-0.5);\n\\end{tikzpicture}}\\\\\n(2) 已知曲线 $\\dfrac{x^2}{a^2}-\\dfrac{y^2}{b^2}=1$($a>0$, $b>0$) 的两条渐近线均和圆 $C: x^2+y^2-6 x+5=0$ 相切, 且双曲线的右焦点为圆 $C$ 的圆心, 则该双曲线的方程为\\bracket{20}.\n\\fourch{$\\dfrac{x^2}{5}-\\dfrac{y^2}{4}=1$}{$\\dfrac{x^2}{4}-\\dfrac{y^2}{5}=1$ }{$\\dfrac{x^2}{3}-\\dfrac{y^2}{6}=1$}{$\\dfrac{x^2}{6}-\\dfrac{y^2}{3}=1$}\n(3) 已知 $F_1(-8,3)$、$F_2(2,3)$ 两点, 动点 $P$ 满足 $|PF_1|-|PF_2|=2 a$. 当 $a$ 分别为 3 和 5 时, 点 $P$ 的轨迹分别为\\bracket{20}.\n\\twoch{双曲线和一条直线}{双曲线和一条射线}{双曲线的一支和一条直线}{双曲线的一支和一条射线}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高二下学期校本作业-解析几何", + "edit": [ + "20240224\t杨懿荔" + ], + "same": [], + "related": [ + "021248" + ], + "remark": "", + "space": "", + "unrelated": [] + }, + "041006": { + "id": "041006", + "content": "已知双曲线方程为 $x^2-\\dfrac{y^2}{4}=1$, 过 $P(1,0)$ 的直线 $l$ 与双曲线只有一个公共点, 则 $l$ 的条数共\\blank{50}条.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高二下学期校本作业-解析几何", + "edit": [ + "20240224\t杨懿荔" + ], + "same": [], + "related": [ + "021257" + ], + "remark": "", + "space": "", + "unrelated": [] + }, + "041007": { + "id": "041007", + "content": "rep[021273,000669,018851,021319,016575] 根据下列条件, 写出抛物线的标准方程:\\\\\n(1) 准线方程是 $x=\\dfrac{1}{4}$;\\\\\n(2) 焦点到准线的距离是 2 ;\\\\\n(3) 过点 $(-3,4)$;\\\\\n(4) 过焦点且与 $x$ 轴垂直的弦长是 16 ;\\\\\n(5) 焦点在直线 $3 x-4 y-12=0$ 上的抛物线标准方程.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高二下学期校本作业-解析几何", + "edit": [ + "20240224\t杨懿荔" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "041008": { + "id": "041008", + "content": "求抛物线 $y=a x^2$ 的焦点坐标和准线方程.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高二下学期校本作业-解析几何", + "edit": [ + "20240224\t杨懿荔" + ], + "same": [], + "related": [ + "010682", + "008924" + ], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "041009": { + "id": "041009", + "content": "动圆 $M$ 经过点 $A(3,0)$ 且与直线 $l: x=-3$ 相切, 则动圆圆心 $M$ 的轨迹方程\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高二下学期校本作业-解析几何", + "edit": [ + "20240224\t杨懿荔" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "041010": { + "id": "041010", + "content": "抛物线 $y^2=2 x$ 上的 $A$、$B$ 两点到焦点 $F$ 的距离之和是 5 , 则线段 $AB$ 的中点的横坐标是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高二下学期校本作业-解析几何", + "edit": [ + "20240224\t杨懿荔" + ], + "same": [], + "related": [ + "010683" + ], + "remark": "", + "space": "", + "unrelated": [] + }, + "041011": { + "id": "041011", + "content": "已知抛物线的顶点在原点, 对称轴为 $x$ 轴, 抛物线上的点 $M(-3, m)$ 到焦点的距离等于 5 , 求抛物线的方程和 $m$ 的值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高二下学期校本作业-解析几何", + "edit": [ + "20240224\t杨懿荔" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "041012": { + "id": "041012", + "content": "一椭圆过点 $A(1,-\\dfrac{\\sqrt{2}}{2})$, 它的中心在抛物线 $y^2=-4 x$ 的顶点上, 且椭圆的左焦点与抛物线的焦点重合, 试求: (1) 抛物线的焦点坐标与准线方程; (2) 椭圆方程; (3) 抛物线与椭圆的交点坐标.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高二下学期校本作业-解析几何", + "edit": [ + "20240224\t杨懿荔" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "041013": { + "id": "041013", + "content": "已知点 $M$ 为抛物线 $y^2=4 x$ 上一动点, $F$ 为抛物线的焦点, 定点 $P(3,1)$, 求 $|MP|+|MF|$ 的最小值, 并求此时点 $M$ 的坐标.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高二下学期校本作业-解析几何", + "edit": [ + "20240224\t杨懿荔" + ], + "same": [], + "related": [ + "002402" + ], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "041014": { + "id": "041014", + "content": "动圆 $M$ 与定直线 $y=2$ 相切, 且与定圆 $C: x^2+(y+3)^2=1$ 相外切, 求动圆圆心 $M$ 的轨迹方程.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高二下学期校本作业-解析几何", + "edit": [ + "20240224\t杨懿荔" + ], + "same": [], + "related": [ + "018936" + ], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "041015": { + "id": "041015", + "content": "点 $P$ 到点 $F(2,0)$ 的距离比它到直线 $x+4=0$ 的距离小 2 , 求动点 $P$ 的轨迹方程.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高二下学期校本作业-解析几何", + "edit": [ + "20240224\t杨懿荔" + ], + "same": [], + "related": [ + "021286" + ], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "041016": { + "id": "041016", + "content": "$P$ 为抛物线 $y^2=2 p x$($p>0$) 上任意一点, $F$ 为焦点, 判断以 $|PF|$ 为直径的圆与 $y$ 轴的位置关系, 并说明理由.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高二下学期校本作业-解析几何", + "edit": [ + "20240224\t杨懿荔" + ], + "same": [], + "related": [ + "021306" + ], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "041017": { + "id": "041017", + "content": "(1) 抛物线 $y^2=8 x$ 的动弦 $AB$ 的长为 16 , 求弦 $AB$ 的中点 $M$ 到 $y$ 轴的最短距离;\\\\\n(2) 抛物线 $y^2=8 x$ 的动弦 $AB$ 的长为 1 , 求弦 $AB$ 的中点 $M$ 到 $y$ 轴的最短距离.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高二下学期校本作业-解析几何", + "edit": [ + "20240224\t杨懿荔" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "041018": { + "id": "041018", + "content": "过抛物线 $x^2=4 y$ 的焦点 $F$ 的直线交抛物线于 $A(x_1, y_1), B(x_2, y_2)$ 两点, 如果 $y_1+y_2=6$, 那么 $|AB|=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高二下学期校本作业-解析几何", + "edit": [ + "20240224\t杨懿荔" + ], + "same": [], + "related": [ + "021307" + ], + "remark": "", + "space": "", + "unrelated": [] + }, + "041019": { + "id": "041019", + "content": "已知抛物线 $y^2=2 p x$($p>0$) 的一条经过焦点的弦被焦点分成长为 $m$、$n$ 的两部分.求证: $\\dfrac{1}{m}+\\dfrac{1}{n}$ 为定值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高二下学期校本作业-解析几何", + "edit": [ + "20240224\t杨懿荔" + ], + "same": [], + "related": [ + "021311" + ], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "041020": { + "id": "041020", + "content": "对于抛物线 $C: y^2=4 x$, 我们称满足 $y_0{}^2<4 x_0$ 的点 $M(x_0, y_0)$ 在抛物线的内部, 若点 $M(x_0, y_0)$ 在抛物线的内部, 则直线 $l: y_0 y=2(x+x_0)$ 与抛物线 $C$\\bracket{20}.\n\\fourch{恰有一个公共点}{恰有二个公共点}{有一个公共点也可能有二个公共点}{没有公共点}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高二下学期校本作业-解析几何", + "edit": [ + "20240224\t杨懿荔" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "041021": { + "id": "041021", + "content": "如图, 过抛物线 $y^2=2 p x$($p>0$) 上一定点 $P(x_0, y_0)$($y_0>0$), 作两条直线分别交抛物线于 $A(x_1, y_1), B(x_2, y_2)$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, line cap = round, line join = round, scale = 1]\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] {$P$};\n\\draw (0.49,-0.7) node [below] {$A$} -- (1.44,1.2) -- (2.89,-1.7) node [below] {$B$} -- cycle;\n\\end{tikzpicture}\n\\end{center}\n(1) 求该抛物线上纵坐标为 $\\dfrac{p}{2}$ 的点到其焦点 $F$ 的距离;\\\\\n(2) 当 $PA$ 与 $PB$ 的斜率存在且倾斜角互补时,求 $\\dfrac{y_1+y_2}{y_0}$ 的值, 并证明直线 $AB$ 的斜率是非零常数.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高二下学期校本作业-解析几何", + "edit": [ + "20240224\t杨懿荔" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "041022": { + "id": "041022", + "content": "方程 $y=-\\sqrt{x^2-2 x+1}$ 的图形是下图中的\\bracket{20}.\n\\fourch{}{}{}{}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高二下学期校本作业-解析几何", + "edit": [ + "20240224\t杨懿荔" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "041023": { + "id": "041023", + "content": "``点 $M(a, b)$ 在曲线 $y^2=x$ 上''是``点 $M(a, b)$ 在曲线 $y=\\sqrt{x}$ 上''的\\blank{50}条件.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高二下学期校本作业-解析几何", + "edit": [ + "20240224\t杨懿荔" + ], + "same": [], + "related": [ + "021333" + ], + "remark": "", + "space": "", + "unrelated": [] + }, + "041024": { + "id": "041024", + "content": "直线 $y=k x-3$ 与曲线 $|x-1|+y=0$ 的公共点最多有\\bracket{20}.\n\\fourch{0 个}{1 个}{2 个}{3 个}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高二下学期校本作业-解析几何", + "edit": [ + "20240224\t杨懿荔" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "041025": { + "id": "041025", + "content": "(1) 已知两条曲线的方程是 $F_1(x, y)=0$ 和 $F_2(x, y)=0$, 它们的交点是 $M(x_0, y_0)$. 求证: 当 $\\lambda \\in \\mathbf{R}$ 时, 方程 $F_1(x, y)+\\lambda F_2(x, y)=0$ 的曲线也经过点 $M(x_0, y_0)$.\\\\\n(2) 已知两条曲线 $x^2+y^2-3 x+y=0$ 和 $3 x^2+3 y^2+4 x-y=0$ 有两个交点, 求经过这两交点的直线的方程.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高二下学期校本作业-解析几何", + "edit": [ + "20240224\t杨懿荔" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "041026": { + "id": "041026", + "content": "求证: 无论 $m$ 取何实数, 方程 $(1+m) x^2+(m^2-1) x+(m^2-2 m-3) y-(2 m^2-m-3)=0$所表示的曲线总是经过一个定点.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高二下学期校本作业-解析几何", + "edit": [ + "20240224\t杨懿荔" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "041027": { + "id": "041027", + "content": "已知方程 $y=\\sqrt{1-x^2}$ 与方程 $y=k x+2$ 有且仅有一个公共解, 求实数 $k$ 的取值范围.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高二下学期校本作业-解析几何", + "edit": [ + "20240224\t杨懿荔" + ], + "same": [], + "related": [ + "019674" + ], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "041028": { + "id": "041028", + "content": "已知椭圆 $\\begin{cases}x=3 \\cos \\theta\\\\y=2 \\sin \\theta\\end{cases}$ ($\\theta$ 为参数), 则\\\\\n(1) $\\theta=\\dfrac{\\pi}{6}$ 时对应的点 $P$ 的坐标是\\blank{50}; (2) 直线 $OP$ 的倾斜角是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高二下学期校本作业-解析几何", + "edit": [ + "20240224\t杨懿荔" + ], + "same": [], + "related": [ + "021344" + ], + "remark": "", + "space": "", + "unrelated": [] + }, + "041029": { + "id": "041029", + "content": "若圆 $C$ 的方程为 $(x-a)^2+(y-b)^2=r^2$, 写出圆 $C$ 的一个参数方程.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高二下学期校本作业-解析几何", + "edit": [ + "20240224\t杨懿荔" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "041030": { + "id": "041030", + "content": "设 $P$、$Q$ 是椭圆 $\\dfrac{x^2}{4}+y^2=1$ 上相异的两点. 设 $A(2,0)$、$B(0,1)$.\n命题甲: 若 $|AP|=|AQ|$ , 则 $P$ 与 $Q$ 关于 $x$ 轴对称;\n命题乙: 若 $|BP|=|BQ|$, 则 $P$ 与 $Q$ 关于 $y$ 轴对称.\n关于这两个命题的真假, 以下四个论述中, 正确的是\\bracket{20}.\n\\fourch{甲和乙都是真命题}{甲是真命题, 乙是假命题}{甲是假命题, 乙是真命题}{甲和乙都是假命题}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高二下学期校本作业-解析几何", + "edit": [ + "20240224\t杨懿荔" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "041031": { + "id": "041031", + "content": "如图, 点 $A$ 是曲线 $y=\\sqrt{x^2+2}$($y \\leq 2$) 上的任意一点, $P(0,-2), Q(0,2)$, 射线 $QA$ 交曲线 $y=\\dfrac{1}{8}x^2$ 于 $B$ 点, $BC$ 垂直于直线 $y=3$, 垂足为点 $C$. 则下列判断:\\\\\n\\textcircled{1} $|AP|-|AQ|$ 为定值 $2 \\sqrt{2}$;\\\\\n\\textcircled{2} $|QB|+|BC|$ 为定值 5 . 其中正确的说法是\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.5]\n\\draw [->] (-5,0) -- (5,0) node [below] {$x$};\n\\draw [->] (0,-3) -- (0,6) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [name path = hline] (-5,3) -- (5,3);\n\\filldraw (0,-2) circle (0.06) node [left] {$P$} coordinate (P);\n\\filldraw (0,2) circle (0.06) node [left] {$Q$} coordinate (Q);\n\\draw [domain = {-sqrt(2)}:{sqrt(2)}, samples = 100] plot (\\x,{sqrt(\\x*\\x+2)});\n\\draw [name path = para, domain = -5:5, samples = 100] plot (\\x, {\\x*\\x/8});\n\\filldraw (1,{sqrt(3)}) circle (0.06) node [above] {$A$} coordinate (A);\n\\path [name path = QA] (Q)--($(Q)!4!(A)$);\n\\draw [name intersections = {of = QA and para, by = B}];\n\\draw (Q)--(B) node [below] {$B$};\n\\path [name path = BC] (B)--++ (0,2);\n\\draw [name intersections = {of = hline and BC, by = C}];;\n\\draw (B)--(C) node [above] {$C$};\n\\end{tikzpicture}\n\\end{center}\n\\fourch{\\textcircled{1}\\textcircled{2}都正确}{\\textcircled{1}\\textcircled{2}都错误}{\\textcircled{1}正确, \\textcircled{2}错误}{\\textcircled{1}都错误, \\textcircled{2}正确}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高二下学期校本作业-解析几何", + "edit": [ + "20240224\t杨懿荔" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "041032": { + "id": "041032", + "content": "已知椭圆 $\\dfrac{x^2}{9}+\\dfrac{y^2}{4}=1$ 的焦点为 $F_1, F_2$, 椭圆上的动点 $P$ 坐标 $(x_0, y_0)$, 且 $\\angle F_1PF_2$ 为锐角, $x_0$ 的取值范围为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高二下学期校本作业-解析几何", + "edit": [ + "20240224\t杨懿荔" + ], + "same": [], + "related": [ + "024135" + ], + "remark": "", + "space": "", + "unrelated": [] + }, + "041033": { + "id": "041033", + "content": "设 $P$ 是双曲线 $x^2-\\dfrac{y^2}{15}=1$ 的右支上一点, 过点 $P$ 分别作圆 $(x+4)^2+y^2=4$ 和 $(x-4)^2+y^2=1$的切线, 切点分别为 $M, N$, 则 $|PM|^2-|PN|^2$ 的最小值为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高二下学期校本作业-解析几何", + "edit": [ + "20240224\t杨懿荔" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "041034": { + "id": "041034", + "content": "点 $P$ 在圆 $C: x^2+(y-2)^2=\\dfrac{1}{9}$ 上移动, 点 $Q$ 在椭圆 $x^2+4 y^2=4$ 上移动, 则 $|PQ|$ 的最大值为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高二下学期校本作业-解析几何", + "edit": [ + "20240224\t杨懿荔" + ], + "same": [], + "related": [ + "002345" + ], + "remark": "", + "space": "", + "unrelated": [] + }, + "041035": { + "id": "041035", + "content": "直线 $x=1$ 上有动点 $P, O$ 为坐标原点, 等腰直角 $\\triangle OPQ$ 中, $\\angle POQ=\\dfrac{\\pi}{2}$, 动点 $Q$ 的轨迹方程为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高二下学期校本作业-解析几何", + "edit": [ + "20240224\t杨懿荔" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "041036": { + "id": "041036", + "content": "设动点 $\\mathrm{A}$ 的轨迹为抛物线 $y^2=4 x$, 点 $B(2,0)$ 为定点. 若线段 $AB$ 的中点为点 $P$, 则点 $P$ 的轨迹方程为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高二下学期校本作业-解析几何", + "edit": [ + "20240224\t杨懿荔" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "041037": { + "id": "041037", + "content": "设 $P$ 是双曲线 $\\dfrac{x^2}{3}-\\dfrac{y^2}{6}=1$ 上的一点, $F_1$、$F_2$ 是该双曲线的左、右焦点. 若 $(\\overrightarrow{F_1P}+\\overrightarrow{F_2P}) \\cdot(\\overrightarrow{F_1P}-\\overrightarrow{F_2P})=72$, 则 $|\\overrightarrow{F_1P}|=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高二下学期校本作业-解析几何", + "edit": [ + "20240224\t杨懿荔" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "041038": { + "id": "041038", + "content": "已知曲线 $C_1:\\begin{cases}x=-4+\\cos t,\\\\y=3+\\sin t,\\end{cases}$ ($t$ 为参数), $C_2:\\begin{cases}x=8 \\cos \\theta,\\\\y=3 \\sin \\theta,\\end{cases}$ ($\\theta$ 为参数).\\\\\n(1) 化 $C_1, C_2$ 的方程为普通方程, 并说明它们分别表示什么曲线;\\\\\n(2) 若 $C_1$ 上的点 $P$ 对应的参数为 $t=\\dfrac{\\pi}{2}$, 点 $Q$ 为 $C_2$ 上的动点, 求 $PQ$ 中点 $M$ 到直线 $C_3:\\begin{cases}x=3+2 t,\\\\y=-2+t\\end{cases}$ ($t$ 为参数) 距离的最小值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高二下学期校本作业-解析几何", + "edit": [ + "20240224\t杨懿荔" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "041039": { + "id": "041039", + "content": "设双曲线 $\\Gamma$ 的方程为 $x^2-\\dfrac{y^2}{4}=1$.\\\\\n(1) 设 $l$ 是经过点 $M(1,1)$ 的直线, 且和 $\\Gamma$ 有且仅有一个公共点, 求 $l$ 的方程;\\\\\n(2) 设 $l_1$ 是 $\\Gamma$ 的一条渐近线, $A$、$B$ 是 $l_1$ 上相异的两点. 若点 $P$ 是 $\\Gamma$ 上的一点, $P$ 关于点 $A$ 的对称点记为 $Q, Q$ 关于点 $B$ 的对称点记为 $T$. 试判断点 $T$ 是否可能在 $\\Gamma$ 上, 并说明理由.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高二下学期校本作业-解析几何", + "edit": [ + "20240224\t杨懿荔" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] } } \ No newline at end of file