题目内容
设
是椭圆
上不关于坐标轴对称的两个点,直线
交
轴于点
(与点
不重合),O为坐标原点.
(1)如果点
是椭圆
的右焦点,线段
的中点在y轴上,求直线AB的方程;
(2)设
为
轴上一点,且
,直线
与椭圆
的另外一个交点为C,证明:点
与点
关于
轴对称.
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824051057191423.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824051057222865.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824051057253396.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824051057269266.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824051057284399.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824051057191423.png)
(1)如果点
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824051057284399.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824051057347396.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824051057362472.png)
(2)设
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824051057393357.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824051057269266.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824051057425723.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824051057440425.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824051057347396.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824051057471309.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824051057487313.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824051057269266.png)
(1)直线
(即
)的方程为
或
;(2)详见解析.
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824051057253396.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824051057362472.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824051057549631.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824051057565648.png)
试题分析:(1)由已知条件推导出点
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824051057471309.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824051057596588.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824051057253396.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824051057362472.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824051057659313.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824051057269266.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824051057690337.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824051057347396.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824051057471309.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824051057487313.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824051057269266.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824051057690337.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824051057440425.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824051057347396.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824051057799300.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824051057799300.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824051057393357.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824051057690337.png)
(1)椭圆
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824051057347396.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824051057877606.png)
因为线段
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824051057362472.png)
所以点
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824051057471309.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824051057924223.png)
因为点
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824051057471309.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824051057347396.png)
将
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824051057971332.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824051057347396.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824051057471309.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824051057596588.png)
所以直线
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824051057253396.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824051057362472.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824051057549631.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824051057565648.png)
(2)设点
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824051057659313.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824051057269266.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824051057690337.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824051057347396.png)
要证点
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824051057471309.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824051057487313.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824051057269266.png)
只要证点
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824051057690337.png)
又因为直线
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824051057440425.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824051057347396.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824051057799300.png)
所以只要证明点
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824051057799300.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824051057393357.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824051057690337.png)
以下给出证明:
由题意,设直线
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824051057253396.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824051058439818.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824051058454616.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824051058470644.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824051058485692.png)
由
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240510585011187.png)
得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240510585321174.png)
所以
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240510585481381.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824051058563984.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240510585951014.png)
在
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824051058610598.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824051058626391.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824051057284399.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824051058673646.png)
由
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824051057425723.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824051057393357.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824051058719709.png)
设直线
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824051058751422.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824051058766470.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824051058782453.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824051058813476.png)
则
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240510588293496.png)
因为
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240510588441149.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240510588601973.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240510588911329.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240510589071846.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240510589221367.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824051058938287.png)
所以
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824051058969672.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824051057799300.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824051057393357.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824051057690337.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824051057471309.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824051057487313.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824051057269266.png)
![](http://thumb.zyjl.cn/images/loading.gif)
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