名校
解题方法
1 . 已知函数
的图像为曲线
,点
、
.
(1)设点
为曲线
上在第一象限内的任意一点,求线段
的长(用
表示);
(2)设点
为曲线
上任意一点,求证:
为常数;
(3)由(2)可知,曲线
为双曲线,请研究双曲线
的性质(从对称性、顶点、渐近线、离心率四个角度进行研究).
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b51e2d48d9ff877ce086d06018e8e1bd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/75f78f2c700c23e7865bac441ea9cf74.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f420fd6dd06fb68755376e5dcfda9c39.png)
(1)设点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7775aa57ca0e62216f3039ed88dceed0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9ac86e1c253297a377e14fb9a1689be8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/79b752f0f189e5d8666daea73e145dff.png)
(2)设点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acc290b44635265137fdf13146b6a6d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c798d5964658916cab26ba10f62485f1.png)
(3)由(2)可知,曲线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
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