题目
题型:不详难度:来源:
设椭圆
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194513-33573.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194513-69680.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194513-40948.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194513-45111.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194514-30963.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194514-96825.gif)
(Ⅰ)证明
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194514-65703.gif)
(Ⅱ)设
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194514-66121.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194514-58600.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194513-45111.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194515-58520.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194515-39539.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194515-51341.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194515-51341.gif)
答案
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194514-65703.gif)
(Ⅱ)点
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194515-51341.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194515-36869.gif)
解析
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194513-40948.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194516-39475.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194516-11926.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194516-28143.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194516-72986.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194517-21611.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194517-75485.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194517-32369.gif)
解得
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194517-60067.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194518-41011.gif)
直线
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194514-30963.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194518-25033.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194518-74141.gif)
由题设,原点
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194513-45111.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194514-30963.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194514-96825.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194518-78076.gif)
将
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194519-42471.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194519-70982.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194514-65703.gif)
证法二:同证法一,得到点
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194517-21611.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194519-31701.gif)
过点
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194513-45111.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194519-81302.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194519-69976.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194520-84691.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194520-14875.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194520-39656.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194520-25640.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194520-49987.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194521-60620.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194521-22465.gif)
所以
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194521-15690.gif)
解得
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194521-26742.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194522-71065.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194522-64210.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194514-65703.gif)
(Ⅱ)解法一:设点
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194515-51341.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194522-56281.gif)
当
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194522-22407.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194523-91080.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194515-58520.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194523-85757.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194515-58520.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194523-24333.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194523-19223.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194523-58744.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194524-35944.gif)
点
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194524-84708.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194524-81114.gif)
将①式代入②式,得
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194524-86011.gif)
整理得
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194525-25720.gif)
于是
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194525-96147.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194525-60153.gif)
由①式得
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194525-93797.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194526-38618.gif)
由
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194514-58600.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194526-31136.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194526-25499.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194526-19920.gif)
将
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194527-91112.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194527-10190.gif)
当
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194527-24796.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194515-58520.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194527-30769.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194524-84708.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194528-25830.gif)
所以
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194528-64783.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194528-81390.gif)
由
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194514-58600.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194526-31136.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194528-38296.gif)
解得
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194529-70474.gif)
这时,点
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194515-51341.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194527-10190.gif)
综上,点
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194515-51341.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194515-36869.gif)
解法二:设点
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194515-51341.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194522-56281.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194515-39539.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194529-98132.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194523-91080.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194515-51341.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194515-58520.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194529-99361.gif)
记
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194529-34685.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194530-54414.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194524-84708.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194530-21769.gif)
由①式得
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194530-25765.gif)
由②式得
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194530-40171.gif)
将③式代入④式得
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194530-36102.gif)
整理得
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194531-34989.gif)
于是
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194531-88915.gif)
由①式得
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194531-62047.gif)
由②式得
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194531-19697.gif)
将⑥式代入⑦式得
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194532-70029.gif)
整理得
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194532-48059.gif)
于是
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194532-35418.gif)
由
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194514-58600.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194526-31136.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194532-66177.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194533-98212.gif)
将
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194529-34685.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194527-10190.gif)
所以,点
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194515-51341.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194515-36869.gif)
核心考点
试题【(本小题满分14分)设椭圆的左、右焦点分别为是椭圆上的一点,,原点到直线的距离为.(Ⅰ)证明;(Ⅱ)设为椭圆上的两个动点,,过原点作直线的垂线,垂足为,求点的轨】;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/20191023/20191023194456-99485.gif)
(Ⅰ)设P点的坐标为
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194456-82884.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194456-48073.gif)
(Ⅱ)求四边形ABCD的面积的最小值.
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194440-94820.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194440-74643.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194440-50030.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194441-18028.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194441-42939.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194441-84631.gif)
在平面直角坐标系xOy中,经过点
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194429-75010.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194429-72115.gif)
(Ⅰ)求k的取值范围;
(Ⅱ)设椭圆与x轴正半轴、y轴正半轴的交点分别为A、B,是否存在常数k,使得向量
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194430-59956.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194430-34824.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194410-18272.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194410-95040.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194410-35694.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194411-85975.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194411-10133.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194411-40457.png)
(I)求在
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194412-91343.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194412-16971.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194411-10133.png)
(II)当
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194412-89982.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194412-69140.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194412-28587.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194354-57188.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194355-98843.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194355-88352.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194355-92239.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194355-47148.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194355-28225.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194356-85777.png)
(I)求面积
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194355-28225.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194356-51328.png)
(II)求面积
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023194355-28225.png)
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