题目
题型:同步题难度:来源:
(2) 过定点Q(1 ,1) 能否作直线l ,使l 与此双曲线相交于Q1,Q2两点,且Q 是弦Q1Q2的中点?若存在,求出直线l的方程;若不存在,请说明理由.
答案
由
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024015839-24040.png)
设直线与双曲线的交点P1(x1,y1),P2(x2,y2).
当2-k2≠0即k2≠2时,
有
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024015839-80334.png)
又点P(2,1)是弦P1P2的中点,
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024015840-55110.png)
当 k=4时
Δ=4k2 (2k-1)2-4(2-k2) (-4k2+4k-3)=56×5>0,
当k2=2即
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024015840-12418.png)
与渐近线的斜率相等,
即
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024015840-45158.png)
综上所述,所求直线方程为y=4x-7.
(2)假设这样的直线l存在,设Q1(x1,y1),Q2(x2,y2),
则有
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024015840-82624.png)
∴x1+x2=2,y1+y2=2,
且
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024015841-55744.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024015841-30095.png)
∴2(x1-x2)(x1+x2)-(y1-y2) (y1+y2)=0,
∴2(x1-x2)-(y1-y2)=0.
若直线Q1Q2⊥QX,则线段Q1Q2的中点不可能是点Q(1,1),
所以直线Q1Q2有斜率,于是
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024015841-48965.png)
∴直线Q1Q2的方程为y-1=2(x-1),即y=2x-1.
由
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024015841-49132.png)
∴Δ=16-24 <0.
这就是说,直线l与双曲线没有公共点,因此这样的直线不存在.
核心考点
试题【已知双曲线方程为2x2-y2=2 . (1) 过定点P(2 ,1) 作直线交双曲线于P1,P2两点,当点P(2 ,1) 是弦P1P2 的中点时,求此直线方程. 】;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/20191024/20191024015824-79053.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024015825-62529.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024015825-38059.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024015825-68088.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024015815-44115.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024015815-68135.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024015815-84701.png)
(I)求双曲线C的方程;
(Ⅱ)设直线l是圆O:x2+y2=2上动点P(x0,y0)(x0y0≠0)处的切线,l与双曲线C交于不同的两点A,B,证明∠AOB的大小为定值.