试题分析:(1)证明:设直线 与曲线 的交点为![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026005039-40910.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026005039-39308.png) ![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026005037-80076.png) ∴ 即:![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026005041-58627.png) ∴ ……………………2分
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026005039-39308.png) 在 上 ∴ ,![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026005040-99768.png) ∴两式相减得: ……………………4分 ∴ 即: ∴曲线 是一个圆 ……………………6分 (2)设直线 与曲线 的交点为 ,
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026005039-39308.png) ![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026005042-36218.png) ∴曲线 是焦点在 轴上的椭圆
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026005039-39308.png) ![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026005038-81005.png) ∴ 即: 将 代入 整理得:
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026005044-39774.png) ∴ , ……………………8分
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026005039-39308.png) 在 上 ∴![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026005044-98213.png) ![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026005044-30544.png) 又![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026005039-39308.png) ![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026005043-75571.png) ∴![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026005045-33719.png) ∴2![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026005045-14000.png) ![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026005045-10620.png) ∴![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026005045-37605.png) ∴![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026005045-85005.png) ∴![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026005046-31107.png) ∴![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026005046-72208.png) ∴ ……………………10分
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026005039-39308.png) ![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026005038-44603.png) ∴![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026005047-59080.png) ∴![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026005047-13118.png)
……………………12分 点评:直线与椭圆相交时,常联立方程利用韦达定理求解关于弦长,中点弦及垂直夹角等问题;求椭圆离心率的题目需要转化出关于 的方程或不等式 |