(1)利用双曲线的基本量的运算和向量的数量积可得
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823212620456329.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823212620472421.png)
。(2)设出直线
l的方程,要注意斜率存在且不为0,直线方程与双曲线方程联立利用判别式和韦达定理,点
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823212620285493.png)
在以线段
AB为直径的圆的外部,就是
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823212620675623.png)
,得
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823212620690528.png)
;
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408232126208621117.png)
可得
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823212620893633.png)
,再转化为横坐标运算,整理得
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408232126210491203.png)
,由
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823212620690528.png)
求出
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823212621158439.png)
。
解:(Ⅰ)由已知
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823212621174582.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823212620191560.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823212621268527.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408232126212831211.png)
,
∵
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823212621299782.png)
,则
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823212621330755.png)
,∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823212621361486.png)
,∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823212621439604.png)
,
解得
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823212620456329.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823212620472421.png)
,∴双曲线的方程为
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823212620300616.png)
.·································· 4分
(Ⅱ)直线
l的斜率存在且不为0,设直线
l:
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823212621767524.png)
,设
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823212621829587.png)
、
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823212621845586.png)
,
由
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823212621860979.png)
得
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823212621892800.png)
,则
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408232126218922093.png)
解得
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823212622001554.png)
. ①································································· 6分
∵点
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823212620285493.png)
在以线段
AB为直径的圆的外部,则
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823212620675623.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408232126221261301.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408232126221411086.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408232126222821165.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408232126222971193.png)
,解得
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823212622453408.png)
. ②
由①、②得实数
k的范围是
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823212620690528.png)
,······················································· 8分
由已知
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408232126208621117.png)
,∵
B在
A、
Q之间,则
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823212620893633.png)
,且
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823212622734369.png)
,
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823212622750832.png)
,则
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823212622765461.png)
,∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408232126227961284.png)
则
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408232126210491203.png)
,··················································· 10分
∵
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823212620690528.png)
,∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823212623062825.png)
,解得
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823212623186502.png)
,又
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823212622734369.png)
,∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823212621158439.png)
.
故
λ的取值范围是
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823212620441399.png)
.···································································· 13分