设双曲线方程
x^2/a^2-y^2/b^2=1
即b^2x^2-a^2y^2-a^2b^2=0
将y=1/3（x-4）带入上式整理得
(9b^2-a^2)x^2+8a^2x-16a^2-9a^2b^2=0
x1+x2=4a^2/(a^2-9b^2)=2*(-2/3)
b^2=4a^2/9
a^2+4a^2/9=c^2
13a^2/9=c^2
c=根13a/3
两准线间距离为9/2
2a^2/c=9/2
a=3根13/4
a^2=117/16
b^2=13/4
双曲线方程
x^2/(117/16)-y^2/(13/4)=1