AB连线过原点,故A点和B点在双曲线的不同分支上,且两坐标关于原点对称,
设P(x0,y0),A(x1,y1),B(-x1,-y1),
PB直线的斜率k1=(y0+y1)/(x0+x1),
PA直线的斜率k2=(y0-y1)/(x0-x1),
k1*k2=(y0^2-y1^2)/(x0^2-x1^2)=2/3,
P点坐标代入方程,x0^2/a^2-y0^2/b^2=1,(1),
A点坐标代入方程,x1^2/a^2-y1^2/b^2=1,(2),
(1)-(2)式,
(x0^2-x1^2)/a^2-(y0^2-y1^2)=0,
b^2/a^2-(y0^2-y2^2)/(x0^2-x2^2)=0,
b^2/a^2-2/3=0,
*c^2-a^2)/a^2=2/3,
(c/a)^2=5/3,
离心率e=c/a=√15/3.