题目
题型:不详难度:来源:
已知圆M:
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023162221-63770.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023162221-51953.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023162221-95250.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023162221-45250.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023162222-83120.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023162222-12390.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023162222-89812.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023162222-44815.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023162222-91002.png)
(Ⅰ) 求点G的轨迹C的方程;
(Ⅱ) 过点(2,0)作直线l,与曲线C交于A,B两点,O是坐标原点,设
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023162223-34074.png)
答案
解:(1)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023162223-86295.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023162223-72960.png)
解析
(1)根据局题中的向量的关系式,运用坐标法表示得到轨迹方程
(2)设直线方程与椭圆的方程联立,然后结合题中的图形的特点和向量的关系式,得到直线关系式,确定直线的存在与否。
解:(1)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023162223-39340.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023162224-22683.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023162224-22683.png)
∴|GN|+|GM|=|MP|=6,故G点的轨迹是以M、N为焦点的椭圆,其长半轴长
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023162224-82293.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023162224-53469.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023162223-86295.png)
(2)因为
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023162224-87378.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023162225-87467.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023162225-58581.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023162225-99900.png)
若l的斜率不存在,直线l的方程为x=2,由
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023162225-11733.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023162225-63609.png)
故l的斜率存在,设l的方程为
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023162226-92055.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023162226-51281.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023162226-48188.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023162226-12925.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023162226-96807.png)
把①、②代入
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023162227-53793.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023162223-72960.png)
核心考点
试题【.(本题满分14分)已知圆M:定点,点为圆上的动点,点在上,点在上,且满足。(Ⅰ) 求点G的轨迹C的方程;(Ⅱ) 过点(2,0)作直线l,与曲线C交于A,B两点】;主要考察你对椭圆的定义与方程等知识点的理解。[详细]
举一反三
已知椭圆
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023162207-35995.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023162207-75452.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023162207-80458.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023162207-16325.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023162208-63767.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023162208-20727.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023162207-80458.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023162207-16325.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023162208-90403.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023162207-16325.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023162208-20727.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023162208-13204.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023162208-92331.jpg)
(1)求椭圆
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023162207-16325.png)
(2)求证:
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023162208-86304.png)
(3)若
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023162208-83564.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023162208-20727.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023162148-99464.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023162148-63909.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023162149-16651.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023162149-72701.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023162149-55311.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023162149-38380.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023162149-72701.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023162149-21870.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023162150-25133.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023162149-55311.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023162150-22804.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023162150-48319.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023162150-18752.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023162151-10355.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023162151-40956.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023162151-73135.png)
(1)求椭圆E的标准方程;
(2)求直线
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023162149-72701.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023162151-17849.png)
(3)求证直线
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023162152-15807.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023162152-43495.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023162143-93453.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023162143-59817.png)
A.![]() | B.![]() | C.![]() | D.![]() |
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023162131-44261.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023162131-51480.png)
(1)求椭圆的方程;
(2)设直线
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023162131-81957.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023162131-15266.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023162118-42569.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023162118-76015.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023162118-25549.png)
(Ⅰ)求椭圆M的方程;
(Ⅱ)设直线l:x=ky+m与椭圆M交手A,B两点,若以AB为直径的圆经过椭圆的右顶点C,求△ABC面积的最大值.
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