已知雙曲線
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824015849004778.png)
的左焦點為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824015849020334.png)
,點
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824015849036289.png)
為雙曲線右支上一點,且
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824015849051396.png)
與圓
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824015849067636.png)
相切于點
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824015849082357.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824015849082399.png)
為線段
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824015849051396.png)
的中點,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824015849114292.png)
為坐標(biāo)原點, 則
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824015849129695.png)
=
試題分析:設(shè)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824015849160340.png)
是雙曲線的右焦點,連接
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824015849176402.png)
,因為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824015849192491.png)
分別是
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824015849207483.png)
的中點,所以
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824015849207850.png)
,所以
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824015849223974.png)
,由雙曲線的定義知,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824015849238623.png)
,故
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824015849129695.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240158492701005.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824015849285224.png)
.
點評:本題考查圓與雙曲線的綜合,解題的關(guān)鍵是正確運用雙曲線的定義,三角形的中位線性質(zhì).
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