题目
题型:不详难度:来源:
![](http://img.shitiku.com.cn/uploads/allimg/20191104/20191104043934-81303.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191104/20191104043934-50142.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191104/20191104043934-99841.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191104/20191104043934-77676.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191104/20191104043935-33019.png)
(1)求椭圆C的方程;
(2)过原点且斜率为
![](http://img.shitiku.com.cn/uploads/allimg/20191104/20191104043935-18507.png)
(3)P是椭圆上异于A1,A2的任一点,直线PA1,PA2,分别交
![](http://img.shitiku.com.cn/uploads/allimg/20191104/20191104043935-74130.png)
答案
![](http://img.shitiku.com.cn/uploads/allimg/20191104/20191104043935-59077.png)
解析
试题分析:(1)因为椭圆C的离心率e=
![](http://img.shitiku.com.cn/uploads/allimg/20191104/20191104043934-99841.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191104/20191104043935-10061.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191104/20191104043936-86855.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191104/20191104043934-77676.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191104/20191104043935-59077.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191104/20191104043936-93525.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191104/20191104043936-25749.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191104/20191104043936-93525.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191104/20191104043936-25749.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191104/20191104043936-26764.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191104/20191104043937-91028.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191104/20191104043937-13861.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191104/20191104043937-19589.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191104/20191104043935-18507.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191104/20191104043937-19589.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191104/20191104043937-91028.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191104/20191104043937-91028.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191104/20191104043937-19589.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191104/20191104043937-18529.png)
试题解析:(1)因为椭圆C的离心率e=
![](http://img.shitiku.com.cn/uploads/allimg/20191104/20191104043934-99841.png)
故设a=2m,c=
![](http://img.shitiku.com.cn/uploads/allimg/20191104/20191104043935-10061.png)
直线A2B2方程为 bx ay ab=0,
即mx 2my 2m2=0.
所以
![](http://img.shitiku.com.cn/uploads/allimg/20191104/20191104043936-86855.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191104/20191104043934-77676.png)
所以 a=2,b=1,椭圆方程为
![](http://img.shitiku.com.cn/uploads/allimg/20191104/20191104043935-59077.png)
由
![](http://img.shitiku.com.cn/uploads/allimg/20191104/20191104043938-84153.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191104/20191104043936-93525.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191104/20191104043936-25749.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191104/20191104043936-93525.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191104/20191104043936-25749.png)
又F2(
![](http://img.shitiku.com.cn/uploads/allimg/20191104/20191104043935-10061.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191104/20191104043938-48622.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191104/20191104043936-93525.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191104/20191104043935-10061.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191104/20191104043936-25749.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191104/20191104043938-29077.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191104/20191104043936-93525.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191104/20191104043935-10061.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191104/20191104043936-25749.png)
所以
![](http://img.shitiku.com.cn/uploads/allimg/20191104/20191104043938-48622.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191104/20191104043938-29077.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191104/20191104043936-93525.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191104/20191104043935-10061.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191104/20191104043936-93525.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191104/20191104043935-10061.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191104/20191104043936-25749.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191104/20191104043936-25749.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191104/20191104043935-18507.png)
所以∠EF2F是锐角. 10分
(3)由(1)可知A1(0,1) A2(0, 1),设P(x0,y0),
直线PA1:y 1=
![](http://img.shitiku.com.cn/uploads/allimg/20191104/20191104043936-26764.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191104/20191104043937-91028.png)
直线PA2:y+1=
![](http://img.shitiku.com.cn/uploads/allimg/20191104/20191104043937-13861.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191104/20191104043937-19589.png)
解法一:设圆G的圆心为(
![](http://img.shitiku.com.cn/uploads/allimg/20191104/20191104043935-18507.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191104/20191104043937-19589.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191104/20191104043937-91028.png)
则r2=[
![](http://img.shitiku.com.cn/uploads/allimg/20191104/20191104043935-18507.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191104/20191104043937-19589.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191104/20191104043937-91028.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191104/20191104043937-19589.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191104/20191104043938-41420.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191104/20191104043937-19589.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191104/20191104043937-91028.png)
OG2=
![](http://img.shitiku.com.cn/uploads/allimg/20191104/20191104043938-41420.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191104/20191104043937-19589.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191104/20191104043937-91028.png)
OT2=OG2 r2=
![](http://img.shitiku.com.cn/uploads/allimg/20191104/20191104043938-41420.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191104/20191104043937-19589.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191104/20191104043937-91028.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191104/20191104043938-41420.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191104/20191104043937-19589.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191104/20191104043937-91028.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191104/20191104043937-18529.png)
而
![](http://img.shitiku.com.cn/uploads/allimg/20191104/20191104043939-54981.png)
所以OT=2,即线段OT的长度为定值2. 16分
解法二:OM·ON=|(
![](http://img.shitiku.com.cn/uploads/allimg/20191104/20191104043937-91028.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191104/20191104043937-19589.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191104/20191104043937-18529.png)
而
![](http://img.shitiku.com.cn/uploads/allimg/20191104/20191104043939-54981.png)
由切割线定理得OT2=OM·ON=4.
所以OT=2,即线段OT的长度为定值2. 16分
核心考点
试题【已知椭圆C:+=1(a>b>0)的离心率e=,椭圆C的上、下顶点分别为A1,A2,左、右顶点分别为B1,B2,左、右焦点分别为F1,F2.原点到直线A2B2的距】;主要考察你对平面向量数量积的意义等知识点的理解。[详细]
举一反三
![](http://img.shitiku.com.cn/uploads/allimg/20191104/20191104043928-36522.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191104/20191104043928-29219.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191104/20191104043928-92623.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191104/20191104043928-36522.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191104/20191104043928-69679.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191104/20191104043928-89366.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191104/20191104043928-72422.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191104/20191104043929-38567.jpg)
A.5 | B.3 | C.![]() | D.![]() |
![](http://img.shitiku.com.cn/uploads/allimg/20191104/20191104043917-47252.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191104/20191104043917-73544.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191104/20191104043917-57909.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191104/20191104043918-97523.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191104/20191104043911-61950.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191104/20191104043911-56139.png)
A.![]() | B.![]() | C.![]() | D.![]() |
![](http://img.shitiku.com.cn/uploads/allimg/20191104/20191104043906-58762.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191104/20191104043906-65084.png)
A.![]() | B.![]() | C.![]() | D.![]() |
![](http://img.shitiku.com.cn/uploads/allimg/20191104/20191104043901-73795.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191104/20191104043901-36770.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191104/20191104043901-26034.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191104/20191104043901-22390.png)
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