Mathematics Question from JEE Main 2023 · 25 January, Shift 1

The vertices of a hyperbola $H$ are $({\pm}6, 0)$ and its eccentricity is $\frac{\sqrt{5}}{2}$. Let $N$ be the normal to $H$ at a point in the first quadrant and parallel to the line $\sqrt{2}x + y = 2\sqrt{2}$. If $d$ is the length of the line segment of $N$ between $H$ and the $y$-axis, then $d^2$ is equal to _____.