If the eccentricity e of the hyperbola x^2/a^2-y^2/b^2=1, passing through (6,4 √ 3 ), satisfies 15(e^2+1)=34 e, then the length of the latus rectum…

If the eccentricity $\displaystyle e$ of the hyperbola $\displaystyle \frac{x^2}{a^2}-\frac{y^2}{b^2}=1$, passing through $\displaystyle (6,4 \sqrt{3})$, satisfies $\displaystyle 15\left(e^2+1\right)=34 e$, then the length of the latus rectum of the hyperbola $\displaystyle \frac{x^2}{b^2}-\frac{y^2}{2\left(a^2+1\right)}=1$ is: