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修改15078题面

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weiye.wang 2023-05-17 22:16:57 +08:00
parent c4906d2d4f
commit edec77c9db
2 changed files with 4 additions and 3 deletions
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2
工具/latex界面修改题目内容.py
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@ -1,6 +1,6 @@
import os,re,json
"""这里编辑题号(列表)后将在vscode中打开窗口, 编辑后保存关闭"""
problems = "15304"
problems = "15078"
editor = "王伟叶"
def generate_number_set(string,dict):

5
题库0.3/Problems.json
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@ -393394,7 +393394,7 @@
},
"015078": {
"id": "015078",
"content": "已知双曲线$C$的中心在坐标原点, 左焦点$F_1$与右焦点$F_2$都在$x$轴上, 离心率为$3$, 过点$F_2$的动直线$l$与双曲线$C$交于点$A$、$B$, 设$\\dfrac{|AF_2| \\cdot|BF_2|}{|AB|^2}=\\lambda$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.2]\n\\draw [->] (-6,0) -- (6,0) node [below] {$x$};\n\\draw [->] (0,-12) -- (0,12) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = {-2*sqrt(30)}:{2*sqrt(30)}, samples = 100] plot ({sqrt(1+\\x*\\x/8)},\\x);\n\\draw [domain = {-2*sqrt(30)}:{2*sqrt(30)}, samples = 100] plot ({-sqrt(1+\\x*\\x/8)},\\x);\n\\filldraw (3,0) circle (0.15) node [below] {$F_2$} coordinate (F_2);\n\\filldraw (-3,0) circle (0.15) node [below] {$F_2$} coordinate (F_2);\n\\end{tikzpicture}\n\\end{center}\n(1) 求双曲线$C$的渐近线方程;\\\\\n(2) 若点$A$、$B$都在双曲线$C$的右支上, 求$\\lambda$的最大值以及$\\lambda$取最大值时$\\angle AF_1B$的正切值; (关于求$\\lambda$的最值, 某学习小组提出了如下的思路可供参考: \\textcircled{1} 利用基本不等式求最值; \\textcircled{2} 设$\\dfrac{|AF_2|}{|AB|}$为$\\mu$, 建立相应数量关系并利用它求最值; \\textcircled{3} 设直线$l$的斜率为$k$, 建立相应数量关系并利用它求最值)\\\\\n(3) 若点$A$在双曲线$C$的左支上(点$A$不是该双曲线的顶点), 且$\\lambda=1$, 求证: $\\triangle AF_1B$是等腰三角形, 且$AB$边的长等于双曲线$C$的实轴长的$2$倍.",
"content": "已知双曲线$C$的中心在坐标原点, 左焦点$F_1$与右焦点$F_2$都在$x$轴上, 离心率为$3$, 过点$F_2$的动直线$l$与双曲线$C$交于点$A$、$B$, 设$\\dfrac{|AF_2| \\cdot|BF_2|}{|AB|^2}=\\lambda$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.2]\n\\draw [->] (-6,0) -- (6,0) node [below] {$x$};\n\\draw [->] (0,-12) -- (0,12) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = {-2*sqrt(30)}:{2*sqrt(30)}, samples = 100] plot ({sqrt(1+\\x*\\x/8)},\\x);\n\\draw [domain = {-2*sqrt(30)}:{2*sqrt(30)}, samples = 100] plot ({-sqrt(1+\\x*\\x/8)},\\x);\n\\filldraw (3,0) circle (0.15) node [below] {$F_2$} coordinate (F_2);\n\\filldraw (-3,0) circle (0.15) node [below] {$F_1$} coordinate (F_1);\n\\end{tikzpicture}\n\\end{center}\n(1) 求双曲线$C$的渐近线方程;\\\\\n(2) 若点$A$、$B$都在双曲线$C$的右支上, 求$\\lambda$的最大值以及$\\lambda$取最大值时$\\angle AF_1B$的正切值; (关于求$\\lambda$的最值, 某学习小组提出了如下的思路可供参考: \\textcircled{1} 利用基本不等式求最值; \\textcircled{2} 设$\\dfrac{|AF_2|}{|AB|}$为$\\mu$, 建立相应数量关系并利用它求最值; \\textcircled{3} 设直线$l$的斜率为$k$, 建立相应数量关系并利用它求最值)\\\\\n(3) 若点$A$在双曲线$C$的左支上(点$A$不是该双曲线的顶点), 且$\\lambda=1$, 求证: $\\triangle AF_1B$是等腰三角形, 且$AB$边的长等于双曲线$C$的实轴长的$2$倍.",
"objs": [],
"tags": [
"第七单元"
@ -393406,7 +393406,8 @@
"usages": [],
"origin": "2023届黄浦区高三二模试题20",
"edit": [
"20230413\t王伟叶"
"20230413\t王伟叶",
"20230517\t王伟叶"
],
"same": [],
"related": [],