题目
题型:不详难度:来源:
(1)求双曲线方程;
(2)设Q是双曲线上一点,且过点F,Q的直线l与y轴交于点M,若
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024022454-46803.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024022454-19286.png)
答案
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024022454-50010.png)
(2)y=±
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024022455-35836.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024022455-93179.png)
解析
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024022455-34138.png)
则有e=
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024022455-53363.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024022455-42030.png)
所以所求的双曲线方程为
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024022454-50010.png)
(2)因为直线l与y轴相交于M且过焦点F(-2,0),
所以l的斜率一定存在,设为k,则l:y=k(x+2),
令x=0,得M(0,2k),
因为
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024022454-46803.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024022454-19286.png)
所以
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024022456-52223.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024022456-86837.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024022456-52223.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024022456-86837.png)
当
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024022456-52223.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024022456-86837.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024022456-20179.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024022456-74572.png)
所以Q的坐标(-
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024022457-39098.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024022457-34672.png)
因为Q在双曲线
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024022454-50010.png)
所以
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024022457-94743.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024022455-35836.png)
所以直线l的方程为y=±
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024022455-35836.png)
当
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024022456-52223.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024022456-86837.png)
同理求得Q(-4,-2k),代入双曲线方程得,
16-
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024022457-94565.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024022455-93179.png)
所以直线l的方程为y=±
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024022455-93179.png)
综上,所求的直线l的方程为y=±
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024022455-35836.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024022455-93179.png)
核心考点
试题【已知双曲线的中心在原点,离心率为2,一个焦点为F(-2,0).(1)求双曲线方程;(2)设Q是双曲线上一点,且过点F,Q的直线l与y轴交于点M,若= 2,求直线】;主要考察你对双曲线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/20191024/20191024022445-86822.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024022445-44322.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024022445-18186.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024022445-43508.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024022445-91128.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024022445-86822.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024022445-21874.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024022446-39975.png)
(1)求轨迹
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024022445-21874.png)
(2)设直线
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024022446-25029.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024022446-81221.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024022446-69991.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024022445-21874.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024022447-92570.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024022447-98586.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024022447-37424.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024022437-65080.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024022437-98291.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024022437-46818.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024022437-27671.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024022437-66207.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024022437-45697.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024022437-36467.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024022437-46009.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024022438-28289.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024022438-49087.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024022438-73787.png)
A.![]() | B.![]() | C.![]() | D.![]() |
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024022426-14770.png)
A.![]() | B.![]() | C.![]() | D.![]() |
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024022421-22809.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024022421-69035.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024022421-67879.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024022421-41301.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024022421-22809.png)
的距离为
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024022421-57392.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024022421-22809.png)
A.2 | B.![]() | C.![]() | D.![]() |
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024022414-17134.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024022415-83570.png)
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