题目
题型:不详难度:来源:
已知椭圆
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026041258-52043.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026041258-50518.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026041258-28249.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026041258-33241.gif)
(Ⅰ)求椭圆
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026041258-65786.gif)
(Ⅱ)设
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026041259-14715.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026041259-86489.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026041259-33207.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026041258-65786.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026041259-48192.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026041259-14090.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026041258-65786.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026041259-14257.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026041259-61721.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026041259-48192.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026041259-25993.gif)
(Ⅲ)在(Ⅱ)的条件下,过点
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026041259-25993.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026041258-65786.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026041259-63299.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026041300-74709.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026041300-15304.gif)
答案
(Ⅰ)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026041300-78901.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026041300-39036.gif)
解析
(Ⅰ)由题意知
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026041300-74881.gif)
所以
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026041300-85849.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026041300-70010.gif)
又因为
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026041300-88237.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026041300-56296.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026041301-37498.gif)
故椭圆
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026041301-71155.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026041300-78901.gif)
(Ⅱ)由题意知直线
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026041301-98431.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026041301-98431.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026041301-40905.gif)
由
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026041301-21302.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026041301-26356.gif)
…………………………………………6分
设点
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026041301-22013.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026041302-93526.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026041302-68400.gif)
直线
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026041302-62532.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026041302-21776.gif)
令
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026041303-91641.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026041303-28318.gif)
将
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026041303-26491.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026041303-78131.gif)
整理,得
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026041304-51526.gif)
由①得
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026041304-64683.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026041304-82221.gif)
整理,得
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026041304-55784.gif)
所以直线
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026041302-62532.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026041304-26405.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026041305-98214.gif)
(Ⅲ)当过点
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026041305-19083.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026041305-91114.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026041305-91114.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026041305-13518.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026041306-87985.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026041306-30150.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026041301-71155.gif)
由
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026041306-78400.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026041306-73835.gif)
易知
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026041306-90161.gif)
所以
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026041307-66637.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026041307-30131.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026041307-27659.gif)
则
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026041307-37792.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026041308-28563.gif)
因为
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026041308-58565.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026041308-49095.gif)
所以
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026041308-44919.gif)
当过点
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026041305-19083.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026041305-91114.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026041304-55784.gif)
解得
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026041308-70578.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026041309-42622.gif)
此时
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026041309-13320.gif)
所以
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026041309-33110.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026041300-39036.gif)
核心考点
试题【(本小题满分13分)已知椭圆的离心率为,以原点为圆心,椭圆的短半轴为半径的圆与直线相切.(Ⅰ)求椭圆的方程;(Ⅱ)设,,是椭圆上关于轴对称的任意两个不同的点,连】;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/20191026/20191026041245-61043.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026041245-66046.gif)
(1)设
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026041245-51823.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026041246-38377.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026041246-20253.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026041246-75147.gif)
(2)在C1上是否存在点P,使得C1在点P处切线与C2相交于两点A、B,且AB的中垂线恰为C1的切线?若存在,求出点P的坐标;若不存在,说明理由。
如图,椭圆
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026041236-40741.gif)
(Ⅰ)求椭圆C的方程;
(Ⅱ)若AB为垂直于x轴的动弦,直线l:x=4与x轴交于点N,直线AF与BN交于点M.
(ⅰ)求证:点M恒在椭圆C上;
(ⅱ)求△AMN面积的最大值.
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026041236-43712.jpg)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026041222-81494.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026041222-50059.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026041222-16619.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026041223-44670.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026041223-55591.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026041223-10557.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026041223-12422.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026041223-99291.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026041222-50059.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026041223-71443.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026041223-41304.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026041223-90887.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026041223-17602.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026041223-99291.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026041223-71443.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026041223-90887.gif)
(1)当
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026041224-53477.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026041223-12422.gif)
(2)当
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026041224-38719.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026041224-40093.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026041224-81329.jpg)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026041212-77700.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026041213-42077.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026041213-34118.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026041207-28916.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026041207-72522.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026041208-70919.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026041208-22906.png)
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