题目
题型:天津高考真题难度:来源:
![](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)
答案
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024014341-75378.gif)
由题设得
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024014341-20996.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024014341-68279.gif)
所以双曲线的方程为
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024014341-78195.gif)
(Ⅱ)设直线l的方程为y=kx+m(k≠0),
点
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024014341-72205.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024014342-18387.gif)
将①式代入②式,得
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024014342-64324.gif)
整理得
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024014342-22074.gif)
此方程有两个不等实根,
于是
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024014342-23804.gif)
整理得
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024014342-10785.gif)
由根与系数的关系可知线段MN的中点坐标
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024014342-57634.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024014343-86481.gif)
从而线段MN的垂直平分线方程为
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024014343-25265.gif)
此直线与x轴,y轴的交点坐标分别为
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024014343-95882.gif)
由题设可得
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024014343-27029.gif)
整理得
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024014343-78816.gif)
将上式代入③式得
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024014344-17963.gif)
整理得
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024014344-76462.gif)
解得
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024014344-91767.gif)
所以k的取值范围是
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024014344-10830.gif)
核心考点
试题【已知中心在原点的双曲线C的一个焦点是F1(-3,0),一条渐近线的方程是x-2y=0,(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】;主要考察你对双曲线等知识点的理解。[详细]
举一反三
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024014335-39764.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024014335-71935.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024014319-14659.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024014319-51627.gif)
(2) 设轨迹E与x轴交于B、D两点,在E上任取一点Q(x1,y1)(y1≠0),直线QB,QD分别交y轴于M,N 两点,求证:以MN为直径的圆过两定点。