解:(1)∵椭圆C
1的方程是
![](http://thumb.1010pic.com/pic5/latex/1164.png)
,
∴a=2,b=1,c=
![](http://thumb.1010pic.com/pic5/latex/21.png)
,
∴双曲线C
2的方程为
![](http://thumb.1010pic.com/pic5/latex/16752.png)
.
(2)直线y=kx+
![](http://thumb.1010pic.com/pic5/latex/53.png)
,双曲线
![](http://thumb.1010pic.com/pic5/latex/16752.png)
两个方程联立,并化简,得:
(1-3k
2)x
2-6
![](http://thumb.1010pic.com/pic5/latex/53.png)
kx-9=0,
∵直线y=kx+
![](http://thumb.1010pic.com/pic5/latex/53.png)
与双曲线C
2恒有两个不同的交点A和B
∴△=(-6
![](http://thumb.1010pic.com/pic5/latex/53.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/pic5/latex/355807.png)
,
![](http://thumb.1010pic.com/pic5/latex/355808.png)
,
∴
![](http://thumb.1010pic.com/pic5/latex/355809.png)
=k
2x
1x
2+
![](http://thumb.1010pic.com/pic5/latex/53.png)
k(x
1+x
2)+2
=
![](http://thumb.1010pic.com/pic5/latex/355810.png)
.
∵
![](http://thumb.1010pic.com/pic5/latex/140235.png)
,
∴-
![](http://thumb.1010pic.com/pic5/latex/21.png)
<k<
![](http://thumb.1010pic.com/pic5/latex/21.png)
,
故k的范围为:-
![](http://thumb.1010pic.com/pic5/latex/21.png)
<k<
![](http://thumb.1010pic.com/pic5/latex/21.png)
.
(3)C
2渐近线为
![](http://thumb.1010pic.com/pic5/latex/355811.png)
,设
![](http://thumb.1010pic.com/pic5/latex/355812.png)
,且p
2>0,p
1<0,
∴P
1P
2的方程为
![](http://thumb.1010pic.com/pic5/latex/355813.png)
,
令y=0,解得P
1P
2与x轴的交点为N(
![](http://thumb.1010pic.com/pic5/latex/355814.png)
,0),
∴
![](http://thumb.1010pic.com/pic5/latex/355815.png)
=-2
![](http://thumb.1010pic.com/pic5/latex/355816.png)
.
∵
![](http://thumb.1010pic.com/pic5/latex/3632.png)
=
![](http://thumb.1010pic.com/pic5/latex/355817.png)
=[
![](http://thumb.1010pic.com/pic5/latex/355818.png)
]
∴p
1p
2=1,
∴△P
1OP
2的面积S=2
![](http://thumb.1010pic.com/pic5/latex/21.png)
.
分析:(1)由椭圆C
1的方程是
![](http://thumb.1010pic.com/pic5/latex/1164.png)
,知a=2,b=1,c=
![](http://thumb.1010pic.com/pic5/latex/21.png)
,由此能求出双曲线C
2的方程.
(2)由直线y=kx+
![](http://thumb.1010pic.com/pic5/latex/53.png)
,双曲线
![](http://thumb.1010pic.com/pic5/latex/16752.png)
两个方程联立,得(1-3k
2)x
2-6
![](http://thumb.1010pic.com/pic5/latex/53.png)
kx-9=0.由直线y=kx+
![](http://thumb.1010pic.com/pic5/latex/53.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/pic5/latex/355807.png)
,
![](http://thumb.1010pic.com/pic5/latex/355808.png)
,
![](http://thumb.1010pic.com/pic5/latex/355809.png)
=
![](http://thumb.1010pic.com/pic5/latex/355810.png)
.由
![](http://thumb.1010pic.com/pic5/latex/140235.png)
,能求出k的范围.
(3)C
2渐近线为
![](http://thumb.1010pic.com/pic5/latex/355811.png)
,设
![](http://thumb.1010pic.com/pic5/latex/355812.png)
,且p
2>0,p
1<0,P
1P
2的方程为
![](http://thumb.1010pic.com/pic5/latex/355813.png)
,令y=0,解得P
1P
2与x轴的交点为N(
![](http://thumb.1010pic.com/pic5/latex/355814.png)
,0),由此能求出△P
1OP
2的面积.
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.