【答案】
分析:(1)由椭圆C
1的方程是
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131101224525479904568/SYS201311012245254799045020_DA/0.png)
,知a=2,b=1,c=
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131101224525479904568/SYS201311012245254799045020_DA/1.png)
,由此能求出双曲线C
2的方程.
(2)由直线y=kx+
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131101224525479904568/SYS201311012245254799045020_DA/2.png)
,双曲线
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131101224525479904568/SYS201311012245254799045020_DA/3.png)
两个方程联立,得(1-3k
2)x
2-6
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131101224525479904568/SYS201311012245254799045020_DA/4.png)
kx-9=0.由直线y=kx+
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131101224525479904568/SYS201311012245254799045020_DA/5.png)
与双曲线C
2恒有两个不同的交点A和B,得k
2+1>0,设A(x
1,y
1),B(x
2,y
2),则有x
1+x
2=
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131101224525479904568/SYS201311012245254799045020_DA/6.png)
,
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131101224525479904568/SYS201311012245254799045020_DA/7.png)
,
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131101224525479904568/SYS201311012245254799045020_DA/8.png)
=
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131101224525479904568/SYS201311012245254799045020_DA/9.png)
.由
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131101224525479904568/SYS201311012245254799045020_DA/10.png)
,能求出k的范围.
(3)C
2渐近线为
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131101224525479904568/SYS201311012245254799045020_DA/11.png)
,设
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131101224525479904568/SYS201311012245254799045020_DA/12.png)
,且p
2>0,p
1<0,P
1P
2的方程为
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131101224525479904568/SYS201311012245254799045020_DA/13.png)
,令y=0,解得P
1P
2与x轴的交点为N(
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131101224525479904568/SYS201311012245254799045020_DA/14.png)
,0),由此能求出△P
1OP
2的面积.
解答:解:(1)∵椭圆C
1的方程是
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131101224525479904568/SYS201311012245254799045020_DA/15.png)
,
∴a=2,b=1,c=
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131101224525479904568/SYS201311012245254799045020_DA/16.png)
,
∴双曲线C
2的方程为
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131101224525479904568/SYS201311012245254799045020_DA/17.png)
.
(2)直线y=kx+
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131101224525479904568/SYS201311012245254799045020_DA/18.png)
,双曲线
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131101224525479904568/SYS201311012245254799045020_DA/19.png)
两个方程联立,并化简,得:
(1-3k
2)x
2-6
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131101224525479904568/SYS201311012245254799045020_DA/20.png)
kx-9=0,
∵直线y=kx+
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131101224525479904568/SYS201311012245254799045020_DA/21.png)
与双曲线C
2恒有两个不同的交点A和B
∴△=(-6
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131101224525479904568/SYS201311012245254799045020_DA/22.png)
k)
2-4×(1-3k
2)×(-9)>0
即k
2+1>0,
设A(x
1,y
1),B(x
2,y
2)
则有x
1+x
2=
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131101224525479904568/SYS201311012245254799045020_DA/23.png)
,
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131101224525479904568/SYS201311012245254799045020_DA/24.png)
,
∴
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131101224525479904568/SYS201311012245254799045020_DA/25.png)
=k
2x
1x
2+
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131101224525479904568/SYS201311012245254799045020_DA/26.png)
k(x
1+x
2)+2
=
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131101224525479904568/SYS201311012245254799045020_DA/27.png)
.
∵
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131101224525479904568/SYS201311012245254799045020_DA/28.png)
,
∴-
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131101224525479904568/SYS201311012245254799045020_DA/29.png)
<k<
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131101224525479904568/SYS201311012245254799045020_DA/30.png)
,
故k的范围为:-
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131101224525479904568/SYS201311012245254799045020_DA/31.png)
<k<
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131101224525479904568/SYS201311012245254799045020_DA/32.png)
.
(3)C
2渐近线为
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131101224525479904568/SYS201311012245254799045020_DA/33.png)
,设
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131101224525479904568/SYS201311012245254799045020_DA/34.png)
,且p
2>0,p
1<0,
∴P
1P
2的方程为
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131101224525479904568/SYS201311012245254799045020_DA/35.png)
,
令y=0,解得P
1P
2与x轴的交点为N(
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131101224525479904568/SYS201311012245254799045020_DA/36.png)
,0),
∴
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131101224525479904568/SYS201311012245254799045020_DA/37.png)
=-2
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131101224525479904568/SYS201311012245254799045020_DA/38.png)
.
∵
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131101224525479904568/SYS201311012245254799045020_DA/39.png)
=
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131101224525479904568/SYS201311012245254799045020_DA/40.png)
=[
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131101224525479904568/SYS201311012245254799045020_DA/41.png)
]
∴p
1p
2=1,
∴△P
1OP
2的面积S=2
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131101224525479904568/SYS201311012245254799045020_DA/42.png)
.
点评:本题主要考查直线与圆锥曲线的综合应用能力,具体涉及到轨迹方程的求法及直线与双曲线的相关知识,解题时要注意合理地进行等价转化.