题目
题型:不详难度:来源:
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025224334-92428.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025224335-75862.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025224335-27240.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025224335-78699.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025224335-26903.png)
(Ⅰ)求椭圆C的方程;
(Ⅱ)求
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025224335-70276.png)
答案
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025224336-96840.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025224336-12658.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025224336-99687.png)
解析
试题分析:(Ⅰ)由题意比例关系先求c,再由离心率求a,从而可求椭圆的方程;(Ⅱ)分直线AB斜率是否存在两种情况讨论:(1)当直线AB垂直于x轴时,易求;(2)当直线AB不垂直于x轴时,先设直线AB的斜率,点M、A、B的坐标,把点A、B坐标代入椭圆方程求k、m之间的关系,再求PQ直线方程,然后与椭圆方程联立方程组,由韦达定理求
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025224335-70276.png)
试题解析:(Ⅰ) 设F2(c,0),则
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025224336-64415.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025224337-20365.png)
因为离心率e=
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025224337-82437.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025224337-89924.png)
所以椭圆C的方程为
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025224336-96840.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025224338-71980.png)
(Ⅱ)当直线AB垂直于x轴时,直线AB方程为x=-
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025224335-78699.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025224338-88499.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025224338-45935.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025224338-17988.png)
当直线AB不垂直于x轴时,设直线AB的斜率为k,M(-
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025224335-78699.png)
由
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025224339-94655.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025224339-82649.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025224339-71102.png)
此时,直线PQ斜率为
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025224339-54241.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025224339-11087.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025224340-59514.png)
联立
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025224340-83058.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025224340-45398.png)
所以
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025224340-91475.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025224341-14694.png)
于是
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025224341-77114.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025224341-91258.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025224341-61588.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025224342-52097.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025224342-83687.png)
令t=1+32m2,1<t<29,则
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025224342-37450.png)
又1<t<29,所以
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025224342-69337.png)
综上,
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025224343-62387.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025224336-12658.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025224336-99687.png)
核心考点
试题【如图,F1,F2是离心率为的椭圆C:(a>b>0)的左、右焦点,直线:x=-将线段F1F2分成两段,其长度之比为1:3.设A,B是C上的两个动点,线段AB的中垂】;主要考察你对曲线与方程的应用等知识点的理解。[详细]
举一反三
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025224323-69840.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025224324-87708.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025224324-91747.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025224324-62748.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025224324-14889.png)
(1)若
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025224325-93683.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025224325-47052.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025224325-71305.png)
(2)若
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025224326-84768.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025224326-15875.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025224317-44112.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025224317-32714.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025224317-93453.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025224318-20700.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025224318-26386.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025224318-28002.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025224318-46501.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025224307-63860.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025224307-49440.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025224307-48988.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025224307-38783.png)
(Ⅰ)求椭圆
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025224307-63860.png)
(Ⅱ)设椭圆的上顶点为
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025224307-22381.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025224307-63860.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025224308-42193.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025224308-29896.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025224308-47882.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025224308-59513.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025224248-28569.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025224248-27178.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025224249-79237.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025224249-75217.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025224249-84714.png)
(1)求点
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025224248-27178.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025224249-83143.png)
(2)设点
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025224250-71659.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025224249-83143.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025224249-83143.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025224250-71659.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025224250-10033.png)
(3)直线
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025224250-25414.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025224250-71659.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025224250-10033.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025224251-93453.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025224250-25414.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025224251-49243.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025224251-45122.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025224238-37401.png)
(Ⅰ)求椭圆C的方程;
(Ⅱ)已知经过定点M(2,0)且斜率不为0的直线
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025224238-36431.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025224238-52513.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025224239-35295.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025224239-68142.png)
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