题目
题型:不详难度:来源:
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023033019-23030.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023033019-32289.png)
(1)求证:直线恒过定点
(2)判断直线被圆
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023033020-47207.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023033020-88412.png)
答案
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023033020-12358.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023033020-63861.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023033020-19255.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023033020-10180.png)
解析
试题分析:(1)证明:直线
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023033020-12358.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023033020-15046.png)
经整理得
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023033020-65030.png)
由
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023033020-88412.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023033020-92898.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023033020-96899.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023033021-54188.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023033021-88501.png)
(2)解:因为直线
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023033020-12358.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023033020-47207.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023033021-36369.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023033020-12358.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023033020-63861.png)
由
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023033021-85008.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023033021-88501.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023033020-63861.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023033021-18035.png)
又直线
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023033020-12358.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023033020-63861.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023033021-41512.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023033020-12358.png)
于是
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023033021-77002.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023033022-73713.png)
最短弦长为
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023033022-28216.png)
综上所述,当直线
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023033020-12358.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023033020-63861.png)
此时
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023033020-19255.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023033020-10180.png)
点评:当直线与圆相交时,圆心到直线的距离、半弦长和半径组成一个直角三角形,这个直角三角形在解题时经常用到.
核心考点
举一反三
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023033008-49612.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023033008-21228.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023033009-88899.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023033001-57041.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023033001-47002.png)
则圆C2的方程为 .
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023032953-77460.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023032953-47045.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023032953-94625.png)
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