题目内容
如图,
为半圆,
为半圆直径,
为半圆圆心,且
,
为线段
的中点,已知
,曲线
过
点,动点
在曲线
上运动且保持
的值不变.
(I)建立适当的平面直角坐标系,求曲线
的方程;
(II)过点
的直线
与曲线
交于
两点,与
所在直线交于
点,
,
证明:
为定值.
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240204577423301.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020457352446.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020457368396.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020457399292.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020457399531.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020457415333.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020457430373.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020457446502.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020457461313.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020457415333.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020457493289.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020457461313.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020457539550.png)
(I)建立适当的平面直角坐标系,求曲线
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020457461313.png)
(II)过点
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020457571309.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020457602280.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020457461313.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020457633550.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020457430373.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020457680318.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020457695771.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020457711731.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020457727479.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240204577423301.png)
(1)
;(2)
.
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020457758635.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020457773556.png)
试题分析:(1)根据题意建立适当的坐标系,以
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020457399292.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020457539550.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020457539550.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240204578361201.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020457851634.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020457695771.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020457711731.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020457633550.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020457914593.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020457929847.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020457945863.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020457961475.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020457992794.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020457773556.png)
试题解析:(1)以
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020457399292.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020458085530.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020458101266.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020458117310.png)
因为动点
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020457493289.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020457461313.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020457539550.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020457415333.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020457461313.png)
所以
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240204582261208.png)
故曲线
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020457461313.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020457399292.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020458288423.png)
则
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020458304684.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020458319539.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020458335352.png)
所以曲线
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020457461313.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020457758635.png)
(2)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240204583825373.png)
解法一:设而不求法
设
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020457851634.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240204584131231.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240204584291242.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020458444801.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020458460982.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020458475987.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020457758635.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240204585071167.png)
化简,得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020457929847.png)
同理由
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020457711731.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020457945863.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020458569488.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020457992794.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020458600573.png)
解法二:联立方程组法
设
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020457851634.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240204586311226.png)
易知
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020457571309.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020458663480.png)
显然直线
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020457602280.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020457602280.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020458725312.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020457602280.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020458756656.png)
将直线
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020457602280.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020457461313.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020458117310.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240204588191020.png)
∴
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020458834866.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020458850889.png)
又 ∵
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020458881769.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020458897979.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020458912652.png)
同理,由
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020458928738.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020458943675.png)
∴
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240204589432067.png)
![](http://thumb.zyjl.cn/images/loading.gif)
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