题目
题型:不详难度:来源:
已知向量
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105030713-31270.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105030713-38952.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105030713-23247.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105030714-52678.gif)
(1)求轨迹E的方程;
(2)证明:存在圆心在原点的圆,使得该圆的任意一条切线与轨迹E恒有两个交点A,B, 且
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105030714-16215.gif)
答案
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105030714-32700.gif)
(2)存在圆心在原点的圆
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105030714-27992.gif)
且
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105030714-63012.gif)
解析
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105030713-23247.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105030713-31270.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105030713-38952.gif)
所以
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105030714-97466.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105030714-96355.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105030714-32700.gif)
(2).设圆心在原点的圆的一条切线为
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105030715-96163.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105030715-42102.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105030715-66498.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105030715-39658.gif)
要使切线与轨迹E恒有两个交点A,B,
则使△=
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105030716-23205.gif)
即
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105030716-34518.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105030716-55101.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105030716-12426.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105030716-84408.gif)
要使
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105030714-63012.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105030717-78614.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105030717-54864.gif)
所以
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105030717-29712.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105030717-23067.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105030716-55101.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105030717-59354.gif)
即
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105030718-68152.gif)
又因为直线
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105030715-96163.gif)
所以圆的半径为
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105030718-95015.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105030718-88761.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105030714-27992.gif)
当切线的斜率不存在时,切线为
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105030719-36195.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105030714-32700.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105030719-33994.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105030719-58614.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105030714-16215.gif)
综上, 存在圆心在原点的圆
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105030714-27992.gif)
且
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105030714-63012.gif)
核心考点
试题【 (本小题满分14分)已知向量, 向量, 且, 动点的轨迹为E.(1)求轨迹E的方程; (2)证明:存在圆心在原点的圆,使得该圆的任意一条切线与轨迹E恒有两个】;主要考察你对平面向量应用举例等知识点的理解。[详细]
举一反三
在
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105030703-57227.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105030703-79239.gif)
(1)判断
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105030703-57227.gif)
(2)若
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105030704-97245.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105030653-47525.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105030653-41382.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105030654-26862.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105030654-10762.gif)
(1)求点
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105030654-32895.gif)
(2)设点
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105030654-32895.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105030654-93279.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105030655-54351.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105030655-36973.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105030655-96803.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105030656-58183.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105030656-85137.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105030656-97694.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105030648-66696.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105030648-18084.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105030648-51888.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105030648-39859.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105030649-81342.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105030649-35172.gif)
A.1 | B.![]() | C.2 | D.![]() |
如图,三定点A(2,1),B(0,-1),C(-2,1); 三 动点D,E,M满足="t," =" t" ,
="t" , t∈[0,1].
(Ⅰ) 求动直线DE斜率的变化范围;
(Ⅱ) 求动点M的轨迹方程.
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105030642-25367.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105030634-57536.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105030635-14608.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105030635-95901.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105030635-62768.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105030635-56116.gif)
A.![]() | B.![]() |
C.![]() | D.3 |
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