渐近线方程为y=（正负根号3）x
b/a=根3
b=根3a
c=（2根号3）/3
a^2+b^2=c^2=4a^2
a^2=1/3
b^2=1
双曲线C方程
x2/(1/3)-y^2=1
以A,B为直径的圆过原点,则OA垂直OB
设A（x1,y1）B(x2,y2)
y1/x1*y2/x2=-1
x1x2+y1y2=0
3x^2-k^2x^2-2kx-2=0
(3-k^2)x^2-2kx-2=0
x1+x2=2k(3-k^2)
x1x2=-2/(3-k^2)
y1y2=(kx1+1)(kx2+1)
=k^2x1x2+k(x1+x2)+1
k^2=1
k=1或 k=-1