The equation $\frac{1}{r} = \frac{1}{8} + \frac{3}{8} \cos \theta$ represents:

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 $S_1P$ at $P_1$. Let $\delta$ be the distance of $P$ from the straight line $SP_1$,and $\beta = S_1P$. Then the greatest integer less than or equal to $\frac{\beta \delta}{9} \sin \frac{\alpha}{2}$ is:

The distance between the tangents of the hyperbola $2x^2 - 3y^2 = 6$ which are perpendicular to the line $x - 2y + 5 = 0$ is

The length of the latus rectum of the hyperbola $9x^2 - 16y^2 - 18x - 32y - 151 = 0$ is

The equation of the tangents to the conic $3x^2 - y^2 = 3$ perpendicular to the line $x + 3y = 2$ is

Find the equation of the hyperbola whose foci are $(6, 5)$ and $(-4, 5)$ and eccentricity is $5/4$.