设等轴双曲线的方程为
x^2/a^2 - y^2/a^2=1; (a>0)
将直线X-2Y=0→x=2y代入得:
(3/a^2)·y^2=1.
则交点处:y=±a/√3.
y1-y2=a/√3-(-a/√3)=2a/√3.
则x=2y=±2a/√3;
x1-x2=2a/√3-(-2a/√3)=4a/√3
则|AB|=√[(y1-y2)^2+(x1-x2)^2]
=2√5·a/√3
即2√5·a /√3=4√15
求得:a=6.
于是,等轴双曲线的方程为:
x^2/6^2 - y^2/6^2=1;
x^2/36 - y^2/36 =1