题目
题型:不详难度:来源:
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154006-88278.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154006-39942.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154006-96195.png)
(1)求椭圆
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154006-39163.png)
(2)是否存在过点
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154006-81059.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154007-23642.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154007-19843.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154007-23642.png)
答案
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154007-81921.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154007-23642.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154007-95022.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154008-60233.png)
解析
试题分析:(1)∵椭圆
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154006-88278.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154006-39942.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154006-96195.png)
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154008-75444.png)
解得:
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154008-14698.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154008-87117.png)
∴椭圆的方程为:
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154007-81921.png)
(2)假设存在过点
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154006-81059.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154007-23642.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154007-19843.png)
若直线
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154007-23642.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154006-81059.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154007-23642.png)
∴直线
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154007-23642.png)
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154009-38763.png)
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154009-47862.png)
∴直线
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154007-23642.png)
∴可设直线
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154007-23642.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154009-40571.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154009-47066.png)
联立
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154010-71649.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154010-20820.png)
∵直线与椭圆相交于不同的两点M、N,
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154010-64842.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154010-52945.png)
设
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154011-64944.png)
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154011-45358.png)
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154011-94072.png)
又
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154007-19843.png)
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154011-50066.png)
化简得
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154012-15131.png)
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154012-80491.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154012-13992.png)
∴直线
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154007-23642.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154012-45215.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154012-53140.png)
∴存在直线
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154007-23642.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154007-95022.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023154008-60233.png)
点评:涉及到直线与圆锥曲线的位置关系时,如果需要设出直线方程,不要忘记考虑直线的斜率是否存在,联立直线与圆锥曲线方程后,不要忘记验证判别式大于零.
核心考点
试题【已知椭圆过点,且离心率.(1)求椭圆的标准方程;(2)是否存在过点的直线交椭圆于不同的两点M、N,且满足(其中点O为坐标原点),若存在,求出直线的方程,若不存在】;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
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023153956-16712.png)
A. 10 | B. 5 | C.![]() | D.![]() |
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023153949-95073.png)
A.![]() | B.![]() | C.![]() | D.![]() |
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023153942-48955.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023153942-84539.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023153943-43816.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023153943-65021.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023153943-12485.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023153943-26117.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023153943-26117.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023153942-48955.png)
A.![]() | B.![]() | C.![]() | D.![]() |
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023153932-86230.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023153932-21547.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023153932-58194.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023153933-67293.png)
(1)求曲线
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023153932-86230.png)
(2)设过(0,-2)的直线
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023153933-30722.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023153932-86230.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023153933-50884.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023153933-51117.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023153933-27447.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023153933-30722.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023153920-73744.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023153920-77867.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023153920-55096.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023153921-90440.png)
(1)求椭圆的标准方程;
(2)四边形ABCD的顶点在椭圆上,且对角线A C、BD过原点O,若
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023153921-37379.png)
(i) 求
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023153921-70333.png)
(ii) 求证:四边形ABCD的面积为定值;
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