题目
题型:不详难度:来源:
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022140650-45272.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022140650-34849.png)
(1)求该椭圆的标准方程;
(2)取垂直于x轴的直线与椭圆相交于不同的两点P、P′,过P、P′作圆心为Q的圆,使椭圆上的其余点均在圆Q外.若PQ⊥P"Q,求圆Q的标准方程.
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022140650-10737.png)
答案
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022140650-92970.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022140651-16380.png)
解析
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022140651-91530.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022140651-12683.png)
∵离心率
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022140651-78582.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022140651-10103.png)
联立①②得:
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022140652-73121.png)
把b2=8代入②得,a2=16.
∴椭圆的标准方程为
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022140650-92970.png)
(2)设Q(t,0),圆Q的半径为r,则圆Q的方程为(x﹣t)2+y2=r2,
不妨取P为第一象限的点,因为PQ⊥P"Q,则P(
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022140652-22229.png)
联立
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022140652-22342.png)
由△=(﹣4t)2﹣4(2t2+16﹣2r2)=0,得t2+r2=8
又P(
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022140652-22229.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022140652-27936.png)
整理得,
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022140653-40420.png)
代入t2+r2=8,得
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022140653-85574.png)
解得:
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022140653-35612.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022140653-73816.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022140654-19918.png)
此时
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022140654-63301.png)
满足椭圆上的其余点均在圆Q外.
由对称性可知,当t<0时,t=﹣
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022140654-85768.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022140653-35612.png)
故所求椭圆方程为
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022140651-16380.png)
核心考点
试题【(2013•重庆)如图,椭圆的中心为原点O,长轴在x轴上,离心率,过左焦点F1作x轴的垂线交椭圆于A、A′两点,|AA′|=4.(1)求该椭圆的标准方程;(2)】;主要考察你对直线方程的几种形式等知识点的理解。[详细]
举一反三
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022140637-61038.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022140638-70012.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022140638-59238.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022140638-46759.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022140638-88359.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022140639-81917.png)
(1)求
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022140639-23456.png)
(2)如果
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022140639-23456.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022140638-46759.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022140638-59238.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022140627-32487.png)
![]() | 3 | -2 | 4 | ![]() |
![]() | ![]() | 0 | -4 | ![]() |
(1)求曲线C1,C2的标准方程;
(2)设直线
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022140628-97614.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022140629-85956.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022140628-97614.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022140628-97614.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022140610-72332.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022140610-66557.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022140610-17696.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022140610-66557.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022140610-60894.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022140610-17696.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022140611-71745.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022140611-41245.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022140611-22801.png)
(1)设
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022140610-60894.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022140611-39197.png)
(2)设
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022140611-75008.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022140610-60894.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022140600-17532.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022140600-81618.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022140600-79780.png)
(1)求方程组没有解的概率;
(2)求以方程组(※)的解为坐标的点落在第四象限的概率..
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022140554-86332.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022140554-98384.png)
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