题目
题型:不详难度:来源:
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023163625-58951.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023163625-74411.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023163625-24226.png)
(1)求椭圆E的方程;
(2)是否存在圆心在原点的圆,使得该圆的任意一条切线与椭圆E恒有两个交点A,B,且OA
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023163625-38670.png)
(3)设直线l与圆
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023163626-47767.png)
答案
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023163626-23572.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023163626-30123.png)
解析
(1)设P (x,y),F1 (–c,0),F2(c,0),其中
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023163626-36548.png)
则
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023163626-95887.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023163627-98578.png)
∴P在A点,x2 + y2最大,∴a2 – c2 = 1,
又
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023163627-10862.png)
(2)由(1)知椭圆方程为
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023163627-33571.png)
①设圆心在原点的圆的一条切线为y = kx + t,
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023163627-63738.png)
解方程组
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023163628-57506.png)
要使切线与椭圆恒有两个交点A,
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023163628-53290.jpg)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023163628-35275.png)
即
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023163628-48666.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023163629-60261.png)
要使
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023163629-11293.png)
所以5t2 – 4k2 – 4 = 0,即5t2 = 4k2 + 4且t2<4k2 + 1,即4k2 + 4<20k2 + 5恒成立.
又因为直线y = kx + t为圆心在原点的圆的一条切线,
所以圆的半径为r =
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023163629-47664.png)
②当切线的斜率不存在时,切线为
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023163629-73876.png)
综上,存在圆心在原点的圆
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023163626-30123.png)
(3)设直线l的方程为y = mx + n,因为直线l与圆C:x2 + y2 = R2 (1<R<2)相切于A1,
由(2)知
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023163630-37116.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023163630-85433.png)
则
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023163630-83965.png)
由①②得
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023163630-57573.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023163631-25381.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023163631-37025.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023163631-16744.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023163631-26733.png)
5
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023163632-97265.png)
因为
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023163632-87497.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023163632-75684.png)
即当
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023163632-71677.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023163632-11724.jpg)
核心考点
试题【已知A、D分别为椭圆E: 的左顶点与上顶点,椭圆的离心率,F1、F2为椭圆的左、右焦点,点P是线段AD上的任一点,且的最大值为1 .(1)求椭圆E的方程;(2】;主要考察你对椭圆的定义与方程等知识点的理解。[详细]
举一反三
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023163617-49549.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023163618-85684.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023163618-76702.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023163618-27103.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023163618-34688.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023163618-58357.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023163619-25832.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023163619-42639.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023163619-37559.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023163618-34688.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023163601-35153.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023163602-67596.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023163602-32183.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023163602-21246.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023163603-99905.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023163603-24518.png)
A.![]() | B.![]() | C.![]() | D.![]() |
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023163557-30787.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023163557-68421.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023163557-86820.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023163557-12537.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023163557-86820.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023163558-11249.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023163558-36355.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023163558-96386.png)
A.一个椭圆,且与![]() |
B.一个椭圆,但与![]() |
C.一个椭圆(去掉长轴的两个端点),且与![]() |
D.一个椭圆(去掉长轴的两个端点),但与![]() |
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023163549-26966.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023163550-92021.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023163550-87918.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023163550-55605.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023163550-27119.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023163550-77558.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023163530-89696.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023163530-46684.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023163530-22494.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023163530-96679.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023163531-77680.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023163531-26484.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023163531-26484.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023163530-96679.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023163531-84301.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023163531-85544.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023163530-96679.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023163531-84301.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023163531-85544.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023163530-96679.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023163531-66049.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023163532-95951.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023163531-66049.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023163532-95951.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023163532-44426.png)
(1)求椭圆
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023163530-96679.png)
(2)证明:点
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023163532-44426.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023163532-35418.png)
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