1 . 如图,曲线
由两个椭圆
和椭圆
组成,当a、b、c成等比数列时,称曲线
为“猫眼曲线”,若猫眼曲线
过点
,且a、b、c的公比为
.
![](https://img.xkw.com/dksih/QBM/2020/12/29/2624460070461440/2627556471996416/STEM/36cb17ee-1f98-4495-8b03-483811e28c7d.png?resizew=322)
(1)求猫眼曲线
的方程;
(2)任作斜率为
且不过原点的直线与该曲线
相交,交椭圆
所得弦AB的中点为M,交椭圆
所得弦CD的中点为N,直线OM、直线ON的斜率分别为
、
,求证:
为与k无关的定值;
(3)设
、
为椭圆
上的两点,直线OP、直线
的斜率分别为
、
,且
,求
的最大值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4b454cdb97c408300b50d945f002c2cb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bb7f0cd04b25e517333647b2580114ac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f8ef77a389b15074e79e324baa70219.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4b454cdb97c408300b50d945f002c2cb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4b454cdb97c408300b50d945f002c2cb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/461b96f2cb34e5d76e4b9f97568025c1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8d5989c84e320b504511f23eeb6e7357.png)
![](https://img.xkw.com/dksih/QBM/2020/12/29/2624460070461440/2627556471996416/STEM/36cb17ee-1f98-4495-8b03-483811e28c7d.png?resizew=322)
(1)求猫眼曲线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4b454cdb97c408300b50d945f002c2cb.png)
(2)任作斜率为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bcb166b53a49e393871bcb14a528792.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4b454cdb97c408300b50d945f002c2cb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4e9a724b59c890095baa5cb73e267c44.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9275bd8ce17fcc4a786510b008414ab0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/80065003fe6677e9a6b4f0dfb9d1db0f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1e4f28029c21c82a381a782e4b16fcbc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/af58317391aa288533c7feee92f52eb1.png)
(3)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5d6f5adf13b4214666292dd64b947741.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e8c774fedfa4c42cd3cb7c234da8baa0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bb7f0cd04b25e517333647b2580114ac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9f0009063fe00277645aff1be6e32471.png)
![](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/151e09f3a383d7811c619eef3a6398cf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a4202e13ac133ddf7aacdbbd70d23dae.png)
您最近一年使用:0次
2 . 已知斜率为
的直线
与椭圆
交于
,
两点,若线段
的中点为
.
(1)证明:
;
(2)设
为
的右焦点,
为
上一点,且
.证明:
,
,
成等差数列.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/279cefeb5c389a37a71e5fd3925f5954.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b3964df6b673a6832a693777d5395890.png)
(1)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/23e402ff27cd20a57b973d94a688e251.png)
(2)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b4ea77b2b480c77f96ee408d58df2ad8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/db36eb152e1ee9a6c138b30ad2ced0aa.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2b32d93c0a0af14ab72fb284ce23c0ba.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8a356309ceefe2eaae58871c32c182e4.png)
您最近一年使用:0次
名校
解题方法
3 . 如图,已知椭圆
,点B是其下顶点,过点B的直线交椭圆C于另一点A(A点在
轴下方),且线段AB的中点E在直线
上.
![](https://img.xkw.com/dksih/QBM/2015/3/31/1572054590701568/1572054596534272/STEM/709902ff189c47ddb3683a316279d417.png?resizew=229)
(1)求直线AB的方程;
(2)若点P为椭圆C上异于A、B的动点,且直线AP,BP分别交直线
于点M、N,证明:
为定值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/219cf5ee39817cd25789e9a687212549.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d77f5191798242b7b9b88a75e17e4425.png)
![](https://img.xkw.com/dksih/QBM/2015/3/31/1572054590701568/1572054596534272/STEM/709902ff189c47ddb3683a316279d417.png?resizew=229)
(1)求直线AB的方程;
(2)若点P为椭圆C上异于A、B的动点,且直线AP,BP分别交直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d77f5191798242b7b9b88a75e17e4425.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/822667ec0ae0e012fe54a9752041088a.png)
您最近一年使用:0次
2016-12-03更新
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