Coordinate Geometry: JEE Main 2026 Mathematics question

If the eccentricity $e$ of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$, passing through $(6,4\sqrt3)$, satisfies $15(e^2+1)=34e$, then the length of the latus rectum of the hyperbola $\frac{x^2}{b^2}-\frac{y^2}{2(a^2+1)}=1$ is: