题目
题型:高考真题难度:来源:
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023122332-61254.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023122332-84164.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023122332-24950.png)
(2)设A,B是椭圆上位于x轴上方的两点,且直线AF1与直线BF2平行,AF2与BF1交于点P。
(i)若AF1-BF2=
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023122332-52026.png)
(ii)求证:PF1+PF2是定值。
答案
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023122332-66872.png)
由点(1,e)在椭圆上,得
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023122333-23909.png)
∴b=1,c2=a2-1
由点(e,
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023122333-50803.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023122333-78314.png)
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023122333-58874.png)
∴a2=2
∴椭圆的方程为
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023122334-30586.png)
(2)解:由(1)得F1(-1,0),F2(1,0),
又∵直线AF1与直线BF2平行,
∴设AF1与BF2的方程分别为x+1=my,x-1=my
设A(x1,y1),B(x2,y2),y1>0,y2>0,
∴由
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023122334-18257.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023122334-73228.png)
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023122334-71590.png)
∴|AF1|=
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023122335-62756.png)
同理|BF2|=
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023122335-97310.png)
(i)由①②得|AF1|-|BF2|=
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023122335-30226.png)
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023122335-15774.png)
∵注意到m>0,
∴m=
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023122336-46660.png)
∴直线AF1的斜率为
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023122336-48601.png)
(ii)证明:∵直线AF1与直线BF2平行,
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023122336-89213.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023122336-30430.png)
由点B在椭圆上知,
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023122336-75451.png)
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023122337-77020.png)
同理
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023122337-96201.png)
∴PF1+PF2=
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023122337-87381.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023122338-39398.png)
由①②得,
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023122338-88084.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023122338-51621.png)
∴PF1+PF2=
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023122338-93361.png)
∴PF1+PF2是定值。
核心考点
试题【如图,在平面直角坐标系xOy中,椭圆(a>b>0)的左、右焦点分别为F1(-c,0),F2(c,0)。已知(1,e)和(e,)都在椭圆上,其中e为椭圆的离心率。】;主要考察你对椭圆等知识点的理解。[详细]
举一反三
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023122322-76349.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023122322-90770.png)
(1)求椭圆的方程;
(2)过点A作两条互相垂直的直线分别交椭圆于M,N两点.求证:直线MN恒过定点P(0,﹣
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023122322-87052.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023122322-12131.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023122308-22871.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023122309-85174.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023122309-61903.png)
(1)求椭圆C的方程;
(2)当△AMN的面积为
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023122309-71796.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023122255-11285.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023122255-16830.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023122255-17669.png)
(1)求椭圆T的方程;
(2)是否存在斜率为
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023122256-70130.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023122256-32104.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023122242-73130.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023122242-36353.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023122243-31614.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023122243-70398.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023122243-11216.png)
(2)在(1)的条件下,过右焦点F2作斜率为k的直线
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023122243-94071.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023122229-55844.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023122229-60400.png)
B(0,b),且△ABF的面积为
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023122229-90903.png)
(Ⅰ)求椭圆E的方程;
(Ⅱ)若
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023122229-43180.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023122229-78734.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023122230-12605.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023122230-83339.png)
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