题目
题型:不详难度:来源:
在平面直角坐标系xOy中,抛物线
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025115333-57876.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025115333-90173.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025115333-46166.jpg)
(Ⅰ)求
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025115333-48361.gif)
(Ⅱ)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025115333-48361.gif)
答案
解:(I)设△AOB的重心为G(x,y),A(x1,y1),B(x2,y2),则
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025115334-89083.gif)
∵OA⊥OB ∴
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025115334-41471.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025115334-35751.gif)
又点A,B在抛物线上,有
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025115334-29479.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025115335-46289.gif)
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025115335-72539.gif)
所以重心为G的轨迹方程为
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025115335-42003.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025115335-31840.gif)
(II)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025115335-81500.gif)
由(I)得
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025115336-36943.gif)
当且仅当
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025115336-17184.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025115336-14196.gif)
所以△AOB的面积存在最小值,存在时求最小值1; …………………13分
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025115337-30984.jpg)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025115337-81549.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025115337-85454.gif)
解析
核心考点
试题【(13分)在平面直角坐标系xOy中,抛物线上异于坐标原点O的两不同动点A、B满足(如图所示).(Ⅰ)求得重心G(即三角形三条中线的交点)的轨迹方程;(Ⅱ)的面积】;主要考察你对抛物线的定义与方程等知识点的理解。[详细]
举一反三
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025115322-36032.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025115322-37675.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025115322-37675.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025115322-72265.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025115322-37675.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025115322-15184.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025115323-48353.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025115323-91866.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025115323-91866.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025115323-94902.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025115323-94902.png)
A.①③ | B.①④ | C.②③ | D.②④ |
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025115302-68565.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025115302-25243.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025115302-75605.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025115303-14350.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025115303-50150.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025115303-62091.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025115303-69161.png)
A.![]() | B.![]() | C.![]() | D.![]() |
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025115248-77060.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025115249-37751.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025115249-23926.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025115249-37751.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025115249-51796.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025115249-23926.png)
A.![]() | B.![]() | C.![]() | D.![]() |
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025115232-60198.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025115232-12955.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025115232-92376.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025115232-16660.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025115232-49622.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025115233-78199.jpg)
(1)求
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025115233-35609.png)
(2)记直线MN的斜率为
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025115233-82912.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025115233-56699.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025115234-95774.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025115226-80323.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025115226-30276.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025115227-27860.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025115227-42078.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025115227-83900.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025115227-14877.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025115228-77950.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025115228-38161.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025115228-51630.png)
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