题目
题型:不详难度:来源:
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024024358-40277.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024024359-47947.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024024359-79696.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024024359-19239.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024024359-50915.jpg)
(1)求双曲线C的方程;
(2)如图,P是双曲线C上一点,A、B两点在双曲线C的两条渐近线上,且分别位于第一、二象限.若
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024024400-93097.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024024400-88924.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024024400-14335.png)
答案
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024024400-84211.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024024400-96072.png)
解析
解:(1)由题意知,双曲线C的顶点(0,a)到渐近线ax-by=0的距离为
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024024359-19239.png)
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024024401-72429.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024024359-19239.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024024401-28219.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024024359-19239.png)
由
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024024401-91333.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024024401-42808.png)
∴双曲线C的方程为
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024024400-84211.png)
(2)由(1)知双曲线C的两条渐近线方程为y=±2x,
设A(m,2m),B(-n,2n),m>0,n>0.
由
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024024400-93097.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024024400-88924.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024024402-24120.png)
将P点坐标代入
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024024400-84211.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024024402-62419.png)
设∠AOB=2θ,∵tan(
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024024402-63202.png)
∴tanθ=
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024024402-21542.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024024402-75092.png)
又|OA|=
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024024403-79102.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024024403-79102.png)
∴S△AOB=
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024024402-21542.png)
=2mn
=
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024024402-21542.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024024403-61295.png)
记S(λ)=
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024024402-21542.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024024403-61295.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024024400-14335.png)
则S′(λ)=
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024024402-21542.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024024403-27549.png)
由S′(λ)=0得λ=1.
又S(1)=2,S
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024024403-72063.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024024403-84064.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024024404-86692.png)
∴当λ=1时,△AOB的面积取得最小值2,当λ=
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024024404-93835.png)
△AOB的面积取得最大值
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024024403-84064.png)
∴△AOB面积的取值范围是
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024024400-96072.png)
核心考点
试题【已知双曲线C的方程为-=1(a>0,b>0),离心率e=,顶点到渐近线的距离为.(1)求双曲线C的方程;(2)如图,P是双曲线C上一点,A、B两点在】;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/20191024/20191024024351-24524.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024024351-73562.png)
(A)
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024024351-24355.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024024344-29129.png)
A.![]() | B.![]() | C.![]() | D.![]() |
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024024327-28306.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024024323-37442.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024024323-76383.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024024323-40315.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024024317-42382.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191024/20191024024317-43645.png)
A.![]() | B.![]() |
C.![]() | D.![]() |
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