题目
题型:不详难度:来源:
OA|=a,|OB|=b(a>2,b>2).
(Ⅰ)求证:(a-2)(b-2)=2;
(Ⅱ)求线段AB中点的轨迹方程;
(Ⅲ)求△AOB面积的最小值.
答案
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022233016-68628.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022233016-49638.png)
解析
(1)由已知中圆C:x2+y2-2x-2y+1=0,直线交x轴、y轴于A、B两点|OA|=a,|OB|=b,我们设以分别求出直线的一般方程,和圆的标准方程,然后根据直线与圆相切,圆心到直线的距离等于半径得到结论;
(2)设线段AB的中点M(x,y),代入(1)的结论,整理后,即可得到答案;
(3)S△AOB=
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022233016-68628.png)
(Ⅰ)证明:圆的标准方程是(x-1)2+(y-1)2=1,设直线方程为
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022233016-87581.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022233016-41213.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022233017-78919.png)
即a2+b2+a2b2+2ab-2a2b-2ab2=a2+b2,即a2b2+2ab-2a2b-2ab2=0,即ab+2-2a-2b=0,即(a-2)(b-2)=2.……………………………4分
(Ⅱ)设AB中点M(x,y),则a=2x,b=2y,代入(a-2)(b-2)=2,得(x-1)(y-1)=
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022233016-68628.png)
(Ⅲ)由(a-2)(b-2)=2得ab+2=2(a+b)≥4
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022233017-35840.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022233017-35840.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022233016-49638.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022233017-35840.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022233016-49638.png)
当且仅当a=b时,ab取最小值6+4
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022233016-49638.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022233016-49638.png)
核心考点
试题【已知与圆C:x2+y2-2x-2y+1=0相切的直线l交x轴,y轴于A,B两点,OA|=a,|OB|=b(a>2,b>2).(Ⅰ)求证:(a-2)(】;主要考察你对点到直线的距离等知识点的理解。[详细]
举一反三
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022233011-38547.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022233011-36868.png)
A.![]() | B.![]() | C.![]() | D.![]() |
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022233006-64931.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022233007-33969.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022233007-60754.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022232958-74508.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022232959-85488.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022232959-46754.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022232959-73685.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022233000-76828.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022233000-25136.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022233000-27988.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022232959-46754.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022233000-25136.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022233000-54785.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022233000-82138.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022233001-46084.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022233001-99484.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022232953-79163.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022232954-96692.png)
圆的标准方程是 .
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022232947-67336.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022232947-92490.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022232947-25459.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022232947-29690.png)
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