题目
题型:湖南省高考真题难度:来源:
已知双曲线x2-y2=2的右焦点为F,过点F的动直线与双曲线相交与A、B两点,点C的坐标是(1,0)。
(1)证明·
为常数;
(2)若动点M满足(其中O为坐标原点),求点M的轨迹方程。
答案
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024014411-33435.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024014411-95965.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024014411-88860.gif)
(1)当AB与x轴垂直时,可设点A,B的坐标分别为
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024014411-57417.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024014411-89798.gif)
此时
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024014411-52014.gif)
当AB不与x轴垂直时,设直线AB的方程是
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024014411-31323.gif)
代入
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024014411-83842.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024014411-62744.gif)
则
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024014412-91238.gif)
所以
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024014412-11152.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024014412-47408.gif)
于是
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024014412-45067.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024014412-40264.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024014412-80238.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024014412-96812.gif)
综上所述,
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024014412-92925.gif)
(2)设
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024014412-77807.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024014413-27254.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024014413-48981.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024014413-77887.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024014413-50947.gif)
由
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024014413-92169.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024014413-24715.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024014413-51369.gif)
于是
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024014414-21511.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024014414-78516.gif)
当AB不与x轴垂直时,
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024014414-49365.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024014414-35269.gif)
又因为A,B两点在双曲线上,
所以
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024014414-16839.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024014415-87708.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024014415-58704.gif)
即
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024014415-12805.gif)
将
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024014415-76568.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024014416-16122.gif)
当AB与x轴垂直时,
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024014416-92604.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024014416-28583.gif)
所以点M的轨迹方程是
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024014416-37445.gif)
核心考点
试题【已知双曲线x2-y2=2的右焦点为F,过点F的动直线与双曲线相交与A、B两点,点C的坐标是(1,0)。(1)证明·为常数;(2)若动点M满足(其中O为坐标原点)】;主要考察你对双曲线等知识点的理解。[详细]
举一反三
(1)证明:动点P的轨迹C为双曲线,并求出C的方程;
(2)过点B作直线交双曲线C的右支于M、N两点,试确定λ的范围,使
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024014349-66749.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024014349-14930.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024014341-25305.gif)
(Ⅰ)求双曲线C的方程;
(Ⅱ)若以k(k≠0)为斜率的直线l与双曲线C相交于两个不同的点M,N,且线段MN的垂直平分线与两坐标轴围成的三角形的面积为
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024014341-77638.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024014335-39764.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024014335-71935.gif)