已知點P為雙曲線
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024190139965212953/SYS201310241901399652129011_ST/0.png)
(a>0,b>0)的右支上一點,F(xiàn)
1、F
2為雙曲線的左、右焦點,使
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024190139965212953/SYS201310241901399652129011_ST/1.png)
(O為坐標原點),且|
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024190139965212953/SYS201310241901399652129011_ST/2.png)
|=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024190139965212953/SYS201310241901399652129011_ST/3.png)
|
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024190139965212953/SYS201310241901399652129011_ST/4.png)
|,則雙曲線離心率為( )
A.
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024190139965212953/SYS201310241901399652129011_ST/5.png)
B.
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024190139965212953/SYS201310241901399652129011_ST/6.png)
C.
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024190139965212953/SYS201310241901399652129011_ST/7.png)
D.
【答案】
分析:先由:∵
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024190139965212953/SYS201310241901399652129011_DA/0.png)
,判斷出∠F
1PF
2=90°,再由
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024190139965212953/SYS201310241901399652129011_DA/1.png)
|=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024190139965212953/SYS201310241901399652129011_DA/2.png)
|
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024190139965212953/SYS201310241901399652129011_DA/3.png)
|,解
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024190139965212953/SYS201310241901399652129011_DA/4.png)
,求出c,由此得到雙曲線離心率.
解答:解:∵
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024190139965212953/SYS201310241901399652129011_DA/5.png)
(O為坐標原點),
∴
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024190139965212953/SYS201310241901399652129011_DA/6.png)
,∴|OF
2|=|OP|=|OF
1|=c,
∴∠F
1PF
2=90°,
設|PF
2|=x,則|PF
1|=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024190139965212953/SYS201310241901399652129011_DA/7.png)
,
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024190139965212953/SYS201310241901399652129011_DA/8.png)
,解得
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024190139965212953/SYS201310241901399652129011_DA/9.png)
,
∴
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024190139965212953/SYS201310241901399652129011_DA/10.png)
=(
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024190139965212953/SYS201310241901399652129011_DA/11.png)
)a,
∴
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024190139965212953/SYS201310241901399652129011_DA/12.png)
.
故選D.
點評:本題考查雙曲線的性質(zhì)和應用,解題時要注意平面向量數(shù)量積的運算.
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