试题与答案

设F1、F2是双曲线x24a-y2a=1的两个焦点,点P在双曲线上,∠F1PF2

题型:选择题

题目:

设F1、F2是双曲线
x2
4a
-
y2
a
=1
的两个焦点,点P在双曲线上,∠F1PF2=90°若△F1PF的面积为1,则a的值是(  )
A.1B.
5
2
C.2D.
5

答案:

双曲线

x2
4a
-
y2
a
=1的实半轴长a1=2
a
,虚半轴长b=
a
,c=
5a

不妨设PF1>PF2,则PF1-PF2=2a1=4

a

F1F22=PF12+PF22,而F1F2=2c=2

5a

得PF12+PF22=(PF1-PF22+2PF1•PF2=20a,

∴PF1•PF2=2a,

S=

1
2
PF1•PF2=a=1

则a的值是1.

故选A.

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