【答案】
分析:根據(jù)題意,設(shè)P(x,y)(x≥1),根據(jù)雙曲線的方程,易得A
1、F
2的坐標,將其代入
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024180430791397088/SYS201310241804307913970069_DA/0.png)
•
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024180430791397088/SYS201310241804307913970069_DA/1.png)
中,可得關(guān)于x、y的關(guān)系式,結(jié)合雙曲線的方程,可得
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024180430791397088/SYS201310241804307913970069_DA/2.png)
•
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024180430791397088/SYS201310241804307913970069_DA/3.png)
=4x
2-x-5=4
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024180430791397088/SYS201310241804307913970069_DA/4.png)
-5-
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024180430791397088/SYS201310241804307913970069_DA/5.png)
,由x的范圍,可得答案.
解答:解:根據(jù)題意,設(shè)P(x,y)(x≥1),
易得A
1(-1,0),F(xiàn)
2(2,0),
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024180430791397088/SYS201310241804307913970069_DA/6.png)
•
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024180430791397088/SYS201310241804307913970069_DA/7.png)
=(-1-x,y)•(2-x,y)=x
2-x-2+y
2,
又x
2-
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024180430791397088/SYS201310241804307913970069_DA/8.png)
=1,故y
2=3(x
2-1),
于是
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024180430791397088/SYS201310241804307913970069_DA/9.png)
•
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024180430791397088/SYS201310241804307913970069_DA/10.png)
=4x
2-x-5=4
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024180430791397088/SYS201310241804307913970069_DA/11.png)
-5-
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024180430791397088/SYS201310241804307913970069_DA/12.png)
,
當(dāng)x=1時,取到最小值-2;
故答案為:-2.
點評:本題考查雙曲線方程的應(yīng)用,涉及最值問題;解題的思路是先設(shè)出變量,表示出要求的表達式,結(jié)合圓錐曲線的方程,將其轉(zhuǎn)化為只含一個變量的關(guān)系式,進而由不等式的性質(zhì)或函數(shù)的最值進行計算.