Let a line $L_{1}$ be tangent to the hyperbola $\frac{x^{2}}{16}-\frac{y^{2}}{4}=1$ and let $L_{2}$ be the line passing through the origin and perpendicular to $L _{1}$. If the locus of the point of intersection of $L_{1}$ and $L_{2}$ is $\left(x^{2}+y^{2}\right)^{2}=$ $\alpha x^{2}+\beta y^{2}$, then $\alpha+\beta$ is equal to

The coordinates of the foci of the rectangular hyperbola $xy = {c^2}$ are

Consider the hyperbola

$\frac{x^2}{100}-\frac{y^2}{64}=1$

with foci at $S$ and $S_1$, where $S$ lies on the positive $x$-axis. Let $P$ be a point on the hyperbola, in the first quadrant. Let $\angle SPS _1=\alpha$, with $\alpha<\frac{\pi}{2}$. The straight line passing through the point $S$ and having the same slope as that of the tangent at $P$ to the hyperbola, intersects the straight line $S_1 P$ at $P_1$. Let $\delta$ be the distance of $P$ from the straight line $SP _1$, and $\beta= S _1 P$. Then the greatest integer less than or equal to $\frac{\beta \delta}{9} \sin \frac{\alpha}{2}$ is. . . . . . . 

Find the equation of the hyperbola satisfying the give conditions: Foci $(\pm 4,\,0),$ the latus rectum is of length $12$

The locus of the foot of the perpendicular from the centre of the hyperbola $xy = c^2$  on a variable tangent is :

The equation of the tangent at the point $(a\sec \theta ,\;b\tan \theta )$ of the conic $\frac{{{x^2}}}{{{a^2}}} - \frac{{{y^2}}}{{{b^2}}} = 1$, is