Coordinate Geometry: JEE Main 2024 Mathematics question

The length of the latus rectum and directrices of hyperbola with eccentricity e are 9 and $x= \pm \frac{4}{\sqrt{3}}$, respectively. Let the line $y-\sqrt{3} x+\sqrt{3}=0$ touch this hyperbola at $\left(x_0, y_0\right)$. If $\mathrm{m}$ is the product of the focal distances of the point $\left(x_0, y_0\right)$, then $4 \mathrm{e}^2+\mathrm{m}$ is equal to _________.