解题方法
1 . 已知圆
与椭圆
相交于点M(0,1),N(0,-1),且椭圆的离心率为
.
![](https://img.xkw.com/dksih/QBM/editorImg/2023/3/25/f271141c-d7af-45f0-87ae-38b9c66492a3.png?resizew=190)
(1)求
的值和椭圆C的方程;
(2)过点M的直线
交圆O和椭圆C分别于A,B两点.
①若
,求直线
的方程;
②设直线NA的斜率为
,直线NB的斜率为
,问:
是否为定值?如果是,求出定值;如果不是,请说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/78daa72a59c90db0c7d94cf76d527623.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ef6b94e42869013745050aba059b58dd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8d5989c84e320b504511f23eeb6e7357.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/3/25/f271141c-d7af-45f0-87ae-38b9c66492a3.png?resizew=190)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/11bc05f41215f9894e11d1df0465751a.png)
(2)过点M的直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
①若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5fce4d81feaf584727ec2cf7a80402ce.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
②设直线NA的斜率为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6defc43285a40f7ccb74c1cc04265eba.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/423b7ae39db552e60ee8b1d27312306f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a2a3f348a942d468f0d77c0dfbb41d87.png)
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2 . 在直角坐标系
中,长为3的线段的两端点
分别在
轴、
轴上滑动,点
为线段
上的点,且满足
.记点
的轨迹为曲线
.
(1)求曲线
的方程;
(2)若点
为曲线
上的两个动点,记
,判断是否存在常数
使得点
到直线
的距离为定值?若存在,求出常数
的值和这个定值;若不存在,请说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7ee31829d0d4d5f779a957d7df8058ab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e52586ca2a3b783bc8092415e2d4bf6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d053b14c8588eee2acbbe44fc37a6886.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/af6d4b1949a539377683191ad3d9ea23.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
(1)求曲线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
(2)若点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e6670479a0083dd2dfd5ad55b47b1ab6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ea8ecfbca35202e0cb1f291423ad7078.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411461db15ee8086332c531e086c40c7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
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