题目
题型:不详难度:来源:
已知中心在原点O,焦点在x轴上的椭圆E过点(1,
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026012417-73858.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026012417-26279.png)
(Ⅰ)求椭圆E的方程;
(Ⅱ)直线x+y+1=0与椭圆E相交于A、B(B在A上方)两点,问是否存在直线l,使l与椭圆相交于C、D(C在D上方)两点且ABCD为平行四边形,若存在,求直线l的方程与平行四边形ABCD的面积;若不存在,请说明理由.
答案
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026012417-57150.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026012417-44004.png)
解析
试题分析:解:(Ⅰ)设椭圆的方程为
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026012418-36257.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026012418-26627.png)
∴椭圆的方程为
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026012417-57150.png)
(Ⅱ)由于直线x+y+1=0过椭圆的左焦点F1(-1,0),且斜率为-1,由对称性可知,存在直线l过椭圆的右焦点F2(1,0),且斜率为-1的直线l:x+y-1=0符合题意.
直线x+y+1=0与直线x+y-1=0的距离为d=
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026012418-66402.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026012418-90926.png)
联立
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026012418-96266.png)
设C(x1,y1),B(x2,y2),则x1+x2=
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026012419-45647.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026012419-45647.png)
|CD|=
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026012419-23900.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026012419-88379.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026012418-90926.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026012419-41931.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026012420-72498.png)
故平行四边形ABCD的面积S=
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026012420-72498.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026012418-90926.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026012417-44004.png)
点评:对于圆锥曲线方程的求解,一般应用待定系数法来得到。同时要采用设而不求的联立方程组的思想,研究直线与圆锥曲线的位置关系。
核心考点
试题【(本题满分12分)已知中心在原点O,焦点在x轴上的椭圆E过点(1,),离心率为.(Ⅰ)求椭圆E的方程;(Ⅱ)直线x+y+1=0与椭圆E相交于A、B(B在A上方)】;主要考察你对曲线与方程的应用等知识点的理解。[详细]
举一反三
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026012409-69114.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026012409-76083.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026012409-19337.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026012409-22067.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026012410-52730.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026012410-10056.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026012410-37324.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026012410-27502.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026012411-40251.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026012358-36952.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026012358-59708.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026012358-59982.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026012358-35876.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026012359-71684.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026012359-66501.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026012359-29892.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026012359-24386.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026012400-65588.png)
(1)求双曲线的方程;
(2)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026012400-51334.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026012352-95435.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026012352-92472.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026012352-69278.png)
A.![]() | B.![]() | C.![]() | D.![]() |
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026012340-76691.png)
过抛物线焦点垂直于对称轴的弦叫做抛物线的通径。如图,已知抛物线
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026012340-45050.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026012340-65983.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026012340-39458.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026012340-83703.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026012340-71173.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026012341-26535.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026012341-77258.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026012341-76282.png)
(1)求出抛物线的通径,证明
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026012341-12398.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026012341-43607.png)
(2)证明:
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026012341-39273.png)
已知椭圆
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026012318-22091.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026012318-16095.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026012318-49108.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026012319-23395.png)
(I)求椭圆的标准方程;
(II)过点
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026012319-92353.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026012319-41170.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026012319-93531.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026012319-39314.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191026/20191026012319-41170.png)
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