题目
题型:不详难度:来源:
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025215749-86518.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025215749-89620.png)
(1)求椭圆C的方程;
(2)点P是椭圆C上除长轴端点外的任一点,过点P作斜率为k的直线l,使得l与椭圆C有且只有一个公共点.设直线PF1,PF2的斜率分别为k1,k2.若k≠0,试证明
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025215749-10460.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025215749-82727.png)
答案
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025215749-74594.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025215749-10460.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025215749-82727.png)
解析
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025215749-86518.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025215750-38220.png)
由题意知
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025215750-73762.png)
又e=
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025215750-98140.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025215749-89620.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025215749-74594.png)
(2)设P(x0,y0)(y0≠0),又F1(-
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025215750-22461.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025215750-22461.png)
知
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025215750-66717.png)
直线l的方程为y-y0=k(x-x0).联立得
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025215751-84192.png)
整理得(1+4k2)x2+8(ky0-k2x0)x+4(
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025215751-73007.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025215751-79390.png)
由题意Δ=0,即(4-
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025215751-79390.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025215751-73007.png)
又
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025215751-64378.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025215751-73007.png)
所以16
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025215751-73007.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025215751-79390.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025215752-15353.png)
所以
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025215749-10460.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025215749-82727.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025215752-56350.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025215752-95749.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025215753-90537.png)
因此
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025215749-10460.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025215749-82727.png)
核心考点
试题【椭圆C:=1(a>b>0)的左、右焦点分别是F1、F2,离心率为,过F1且垂直于x轴的直线被椭圆C截得的线段长为1.(1)求椭圆C的方程;(2)点P是椭圆C上除】;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/20191025/20191025215739-79094.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025215740-54811.png)
(1)求椭圆E的方程;
(2)若圆x2+y2=
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025215740-20324.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025215729-60196.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025215729-60196.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025215729-36589.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025215729-46614.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025215730-97110.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025215730-88657.png)
(1)求点P的轨迹方程;
(2)过点(
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025215730-57175.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025215716-68827.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025215716-53629.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025215716-68827.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025215716-53629.png)
点(1,
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025215717-13380.png)
(1)求椭圆C的方程;
(2)过
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025215716-68827.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025215717-35967.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025215717-12735.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025215716-53629.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025215717-51206.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025215716-53629.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025215717-35967.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025215709-86862.jpg)
(1)求抛物线C的方程;
(2)过点F作直线交抛物线C于A,B两点,若直线AO,BO分别交直线l:y=x-2于M,N两点,求|MN|的最小值.
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025215700-53282.png)
(1)求抛物线C的方程;
(2)当点P(x0,y0)为直线l上的定点时,求直线AB的方程;
(3)当点P在直线l上移动时,求|AF|·|BF|的最小值.
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