Let the foci of the ellipse $\frac{x^{2}}{16}+\frac{y^{2}}{7}=1$ and the hyperbola $\frac{ x ^{2}}{144}-\frac{ y ^{2}}{\alpha}=\frac{1}{25}$ coincide. Then the length of the latus rectum of the hyperbola is:-

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. . . . . . . 

For the Hyperbola ${x^2}{\sec ^2}\theta - {y^2}cose{c^2}\theta = 1$ which of the following remains constant when $\theta $ varies $= ?$

Centre of hyperbola $9{x^2} - 16{y^2} + 18x + 32y - 151 = 0$ is

The eccentricity of the hyperbola conjugate to ${x^2} - 3{y^2} = 2x + 8$ is

The equation of the hyperbola whose conjugate axis is $5$ and the distance between the foci is $13$, is