解:(Ⅰ)设双曲线方程为
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823172902343507.gif)
(
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823172902358239.gif)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823172902374241.gif)
),
则
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823172902389308.gif)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823172902421537.gif)
,∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823172902436237.gif)
.------------------------(2分)
又
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823172902452536.gif)
在双曲线上,∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823172902467680.gif)
.
联立①②③,解得
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823172902483292.gif)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823172902499215.gif)
.∴双曲线方程为
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823172902514393.gif)
.--------(4分)
注:对点
M用第二定义,得
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823172902530253.gif)
,可简化计算.
(Ⅱ)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823172902561306.gif)
,设
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823172902577414.gif)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823172902592425.gif)
,
m:
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823172902608433.gif)
,则
由
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823172902639437.gif)
,得
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823172902670471.gif)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823172902686378.gif)
.--------------------(6分)
由
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823172902701628.gif)
,得
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823172902733620.gif)
.
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823172902748565.gif)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823172902764530.gif)
.
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823172902795747.gif)
.
由
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823172902686378.gif)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823172902857528.gif)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823172902764530.gif)
,---------------------(8分)
消去
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823172903076200.gif)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823172903107203.gif)
,
得
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823172903123706.gif)
.------------------------(9分)
∵
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823172903279241.gif)
,函数
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823172903294535.gif)
在
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823172903310273.gif)
上单调递增,
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823172903325661.gif)
,∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823172903341425.gif)
.------------------------(10分)
又直线
m与双曲线的两支相交,即方程
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823172902733620.gif)
两根同号,
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823172903403250.gif)
.------------------------------------------------(11分)
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823172903466470.gif)
,故
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823172903481595.gif)
.------------------------(12分)