题目
题型:不详难度:来源:
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025232858-72193.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025232858-32022.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025232858-67927.png)
(I)求椭圆C的标准方程;
(II)直线x=2与椭圆C交于P、Q两点,A、B是椭圆O上位于直线PQ两侧的动点,且直线AB的斜率为
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025232858-72193.png)
①求四边形APBQ面积的最大值;
②设直线PA的斜率为
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025232859-85267.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025232859-74772.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025232859-85267.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025232859-74772.png)
答案
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025232859-40433.png)
(2)故当
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025232859-36518.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025232900-72604.png)
解析
试题分析:解:(Ⅰ)设椭圆C的方程为
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025232900-14865.png)
由已知b=
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025232900-44244.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025232901-51990.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025232901-74116.png)
所以,椭圆C的方程为
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025232859-40433.png)
(Ⅱ)①由(Ⅰ)可求得点P、Q的坐标为
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025232901-96512.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025232901-27533.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025232902-75606.png)
设A
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025232902-73087.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025232902-71960.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025232902-18482.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025232859-40433.png)
得:
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025232903-65156.png)
由△>0,解得
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025232903-22671.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025232903-87667.png)
四边形APBQ的面积
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025232903-94259.png)
故当
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025232859-36518.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025232904-61586.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025232904-52852.png)
则
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025232904-30597.png)
=
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025232905-18863.png)
=
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025232905-86338.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025232903-87667.png)
可得
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025232905-33750.png)
所以
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025232900-72604.png)
点评:主要是考查了直线与椭圆的位置关系的运用,属于中档题。
核心考点
试题【已知椭圆C的中心在原点,焦点在x轴上,离心率为,短轴长为4.(I)求椭圆C的标准方程;(II)直线x=2与椭圆C交于P、Q两点,A、B是椭圆O上位于直线PQ两侧】;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/20191025/20191025232850-41478.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025232850-93457.png)
A.
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025232850-32773.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025232850-81119.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025232851-90598.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025232851-96658.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025232846-76272.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025232830-40251.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025232830-40251.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025232830-92773.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025232830-39879.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025232831-16600.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025232831-54829.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025232831-26937.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025232831-42111.png)
(1)求动点P所在曲线C的方程;
(2)直线
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025232832-41307.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025232832-31609.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025232832-36227.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025232833-51335.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025232833-65885.png)
(3)记
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025232833-79138.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025232833-82255.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025232834-13007.png)
(A、B、
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025232832-36227.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025232834-85684.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025232834-19258.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025232824-49874.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025232824-81219.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025232825-94984.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025232825-88672.jpg)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025232814-44627.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025232814-20300.png)
Ⅰ)求椭圆C的标准方程;
Ⅱ)若过点B(2,0)的直线
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025232815-47830.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025232815-49518.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025232815-49518.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025232815-20851.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025232815-47830.png)
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