雙曲線kx2-y2=1,右焦點為F,斜率大于0的漸近線為l,l與右準線交于A,F(xiàn)A與左準線交于B,與雙曲線左支交于C,若B為AC的中點,求雙曲線方程.
【答案】
分析:由題設(shè)條件求出A(
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023214638843102243/SYS201310232146388431022014_DA/0.png)
,
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023214638843102243/SYS201310232146388431022014_DA/1.png)
),B(-
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023214638843102243/SYS201310232146388431022014_DA/2.png)
,
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023214638843102243/SYS201310232146388431022014_DA/3.png)
).由B是AC中點,知x
C=2x
B-x
A=-
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023214638843102243/SYS201310232146388431022014_DA/4.png)
,y
C=2y
B-y
A=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023214638843102243/SYS201310232146388431022014_DA/5.png)
.
將x
C、y
C代入方程kx
2-y
2=1,得k
2c
4-10kc
2+25=0.求出k的值,從而得到雙曲線方程.
解答:解:由題意k>0,c=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023214638843102243/SYS201310232146388431022014_DA/6.png)
,
漸近線方程l為y=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023214638843102243/SYS201310232146388431022014_DA/7.png)
x,
準線方程為x=±
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023214638843102243/SYS201310232146388431022014_DA/8.png)
,于是A(
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023214638843102243/SYS201310232146388431022014_DA/9.png)
,
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023214638843102243/SYS201310232146388431022014_DA/10.png)
),
直線FA的方程為y=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023214638843102243/SYS201310232146388431022014_DA/11.png)
,
于是B(-
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023214638843102243/SYS201310232146388431022014_DA/12.png)
,
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023214638843102243/SYS201310232146388431022014_DA/13.png)
).
由B是AC中點,則x
C=2x
B-x
A=-
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023214638843102243/SYS201310232146388431022014_DA/14.png)
,
y
C=2y
B-y
A=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023214638843102243/SYS201310232146388431022014_DA/15.png)
.
將x
C、y
C代入方程kx
2-y
2=1,得
k
2c
4-10kc
2+25=0.
解得k(1+
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023214638843102243/SYS201310232146388431022014_DA/16.png)
)=5,則k=4.
所以雙曲線方程為:4x
2-y
2=1.
點評:本題考查雙曲線的性質(zhì)和綜合運用,解題時要認真審題,仔細解答.