The equation of the tangent to the hyperbola $4{y^2} = {x^2} – 1$ at the point $(1, 0)$ is

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 $…………$.

The distance between the directrices of the hyperbola $x = 8\sec \theta ,\;\;y = 8\tan \theta $ is

The equation of the normal to the hyperbola $\frac{{{x^2}}}{{16}} – \frac{{{y^2}}}{9} = 1$ at $( – 4,\;0)$ is

If line $ax$ + $by$ = $1$ is normal to the hyperbola $\frac{{{x^2}}}{{{p^2}}} – \frac{{{y^2}}}{{{q^2}}} = 1$ then $\frac{{{p^2}}}{{{a^2}}} – \frac{{{q^2}}}{{{b^2}}} = 1$ is equal to (where $a$,$b$,$p$, $q \in {R^ + })$-