Step 1:
$4x^2-9y^2=576$
The above equation is divided by $576$
$\large\frac{x^2}{144}-\frac{y^2}{64}$$=1$
$a^2=144,b^2=64$
$\Rightarrow a=12,b=8$
Step 2:
Eccentricity $e=\sqrt{1+\large\frac{b^2}{a^2}}=\sqrt{1+\large\frac{64}{144}}$
$\Rightarrow \sqrt{1+\large\frac{4}{9}}=\large\frac{\sqrt{13}}{3}$
Step 3:
Directrices : $x=\pm \large\frac{a}{e}$
$x=\pm \large\frac{12}{\sqrt{13}/3}$
$x=\pm \large\frac{36}{\sqrt{13}}$
Step 4:
Latus rectum : $x=\pm ae$
$x=\pm 12\times \large\frac{\sqrt{13}}{3}$
$x=\pm 4\sqrt {13}$
Step 5:
Length of $LR=\large\frac{2b^2}{a}$
$\Rightarrow \large\frac{2\times 64}{12}=\large\frac{32}{3}$