在平面直角坐標(biāo)系中,已知定點(diǎn)A(-2,0)、B(2,0),異于A、B兩點(diǎn)的動點(diǎn)P滿足
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021710678594.png)
,其中k
1、k
2分別表示直線AP、BP的斜率.
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240217107094243.jpg)
(Ⅰ)求動點(diǎn)P的軌跡E的方程;
(Ⅱ)若N是直線x=2上異于點(diǎn)B的任意一點(diǎn),直線AN與(I)中軌跡E交予點(diǎn)Q,設(shè)直線QB與以NB為直徑的圓的一個(gè)交點(diǎn)為M(異于點(diǎn)B),點(diǎn)C(1,0),求證:|CM|·|CN| 為定值.
(Ⅰ)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021710741889.png)
;(Ⅱ)詳見解析.
試題分析:(Ⅰ)根據(jù)斜率公式,有斜率乘積等于
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021710756327.png)
整理即得,注意
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021710772432.png)
;(Ⅱ)設(shè)直線
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021710787429.png)
的方程,與橢圓方程組成方程組,消去
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021710819310.png)
,由韋達(dá)定理求點(diǎn)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021710834333.png)
的坐標(biāo),根據(jù)直線
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021710850406.png)
與以
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021710865442.png)
為直徑的圓的另一個(gè)交點(diǎn)為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021710881399.png)
,得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021710897716.png)
,從而得到直線
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021710912513.png)
的方程,確定恒過的定點(diǎn).證明
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021710943617.png)
三點(diǎn)共線,又
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021710959391.png)
是以
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021710865442.png)
為直徑的圓的切線,由切割線定理可知,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021710990939.png)
,即為定值.
試題解析:(Ⅰ)設(shè)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021711006568.png)
,由
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021711021582.png)
得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021711053765.png)
,其中
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021711068435.png)
,
整理得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021711084289.png)
點(diǎn)的軌跡方程為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021710741889.png)
. (4分)
(Ⅱ)設(shè)點(diǎn)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021711131815.png)
,則直線
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021710787429.png)
的方程為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021711177811.png)
,
解方程組
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240217111931204.png)
,消去
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021710819310.png)
得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240217112241028.png)
,
設(shè)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021711240636.png)
,則
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021711255951.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021711287195.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240217113021105.png)
,
從而
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240217113181095.png)
,又
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021711333547.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240217113491277.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021711365235.png)
直線
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021710850406.png)
與以
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021710865442.png)
為直徑的圓的另一個(gè)交點(diǎn)為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021710881399.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021711287195.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021710897716.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021711474542.png)
方程為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021711630709.png)
,即
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021711645650.png)
,過定點(diǎn)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021711661516.png)
, (9分)
定值證法一:即
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021710943617.png)
三點(diǎn)共線,又
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021710959391.png)
是以
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021710865442.png)
為直徑的圓的切線,由切割線定理可知,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021710990939.png)
,為定值. (12分)
定值證法二:直線
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021711755406.png)
:
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021711786780.png)
,直線
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021711801432.png)
:
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021711645650.png)
,
聯(lián)立得,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021711833814.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240217118791813.png)
,為定值. (12分)
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![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824015817539362.png)
、
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824015817539366.png)
、
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、
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,
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