题目
题型:不详难度:来源:
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025215257-38199.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025215257-81900.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025215257-28149.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025215258-92916.png)
(1)求椭圆C的方程;
(2)求
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025215258-56982.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025215258-49335.png)
(3)若B点关于x轴的对称点是E,证明:直线AE与x轴相交于定点.
答案
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025215259-71036.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025215259-70035.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025215259-14073.png)
解析
(1)解:由题意知e=
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025215259-41236.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025215257-28149.png)
∴e2=
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025215300-35056.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025215300-82669.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025215300-94578.png)
即a2=
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025215300-36744.png)
又b=
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025215301-17058.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025215301-51611.png)
∴b2=3,a2=4,
故椭圆的方程为
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025215259-71036.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025215259-70035.png)
(2)解:由题意知直线l的斜率存在,
设直线l的方程为y=k(x-4).
由
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025215301-30932.png)
得(4k2+3)x2-32k2x+64k2-12=0.
由Δ=(-32k2)2-4(4k2+3)(64k2-12)>0,
得k2<
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025215300-94578.png)
设A(x1,y1),B(x2,y2),
则
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025215301-60514.png)
∴y1y2=k2(x1-4)(x2-4)=k2x1x2-4k2(x1+x2)+16k2,
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025215258-56982.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025215258-49335.png)
=(1+k2)·
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025215302-57719.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025215302-10610.png)
=25-
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025215302-97438.png)
∵0≤k2<
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025215300-94578.png)
∴-
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025215303-50517.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025215302-97438.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025215303-12772.png)
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025215258-56982.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025215258-49335.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025215259-14073.png)
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025215258-56982.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025215258-49335.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025215259-14073.png)
(3)证明:∵B、E两点关于x轴对称,
∴E(x2,-y2).
直线AE的方程为y-y1=
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025215303-68180.png)
令y=0得x=x1-
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025215303-21355.png)
又y1=k(x1-4),y2=k(x2-4),
∴x=
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025215304-38204.png)
将(*)式代入得,x=1,
∴直线AE与x轴交于定点(1,0).
核心考点
试题【已知椭圆C:+=1(a>b>0)的离心率为,以原点为圆心,椭圆的短半轴为半径的圆与直线x-y+=0相切,过点P(4,0)且不垂直于x轴直线l与椭圆C】;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/20191025/20191025215247-88265.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025215248-73435.png)
(1)求椭圆C的标准方程及离心率;
(2)O为坐标原点,P是直线F1A上的一个动点,求|PF2|+|PO|的最小值,并求出此时点P的坐标.
①设A、B为两个定点,k为非零常数,
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025215240-66277.png)
②过定圆C上一定点A作圆的动弦AB,O为坐标原点,若
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025215240-52706.png)
③双曲线
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025215241-71751.png)
④在平面内,到定点
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025215241-77499.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025215241-50297.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025215225-17731.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025215226-84370.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025215226-86184.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025215226-37394.png)
(1)求椭圆
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025215225-17731.png)
(2)若直线
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025215227-54371.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025215225-17731.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025215227-65132.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025215227-93978.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025215227-53743.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025215200-80899.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025215201-55456.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025215201-46348.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025215201-78746.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025215202-99412.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025215202-17806.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025215202-74386.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025215202-61011.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025215203-83317.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025215203-73015.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025215203-46275.png)
A.![]() | B.![]() |
C.![]() | D.![]() |
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025215153-75406.png)
(1)求椭圆的标准方程;
(2)求以OM为直径且被直线3x-4y-5=0截得的弦长为2的圆的方程;
(3)设F是椭圆的右焦点,过点F作OM的垂线与以OM为直径的圆交于点N,求证:线段ON的长为定值,并求出这个定值.
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