本题主要考查直线、圆、椭圆等基础知识,考查推理论证能力、运算求解能力,考查数形结合思想、化归与转化思想
(Ⅰ)依题意可知,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823141914291109.gif)
P与
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823141914291109.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823141914323213.gif)
外切、
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823141914291109.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823141914011215.gif)
内切. 设
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823141914291109.gif)
P的半径为
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823141914401185.gif)
,则
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823141914572548.gif)
-----------------------------------2分
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823141914588128.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823141914603561.gif)
, 2
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823141914619192.gif)
=4,2
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823141914650182.gif)
=
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823141914666281.gif)
=2
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823141914588128.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823141914619192.gif)
=2,c="1" , 椭圆C的方程为
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823141914245245.gif)
+
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823141914260352.gif)
="1 " -----------------------
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/2014082314191488465.gif)
-4分
(Ⅱ)直线AB:y=k(x-1),由
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823141914931142.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823141914993656.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823141914931142.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823141915025755.gif)
,令A
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823141915056576.gif)
,则
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823141915071620.gif)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823141915087622.gif)
, ------------------------------------6分
∵
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823141915103711.gif)
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823141915212729.gif)
,
∵2=
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823141915227288.gif)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823141915243441.gif)
, ------------------------------------8分
2+
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823141915259980.gif)
=
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408231419152901757.gif)
=
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823141915305716.gif)
, ∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823141914276555.gif)
. -----------------------10分
(Ⅲ)设Q(
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823141915337362.gif)
),M(
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823141915352347.gif)
),N(
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823141915368340.gif)
)
则切线QM:
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823141915383507.gif)
切线QN:
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823141915399528.gif)
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823141915415920.gif)
∴M、N在直线
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823141915430532.gif)
上
∴ 直线MN:
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823141915430532.gif)
------------------------------------12分
又
∴直线MN必过定点(
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823141914276276.gif)
). ------------------------------------13分