If the length of the transverse and conjugate axes of a hyperbola be $8$ and $6$ respectively, then the difference focal distances of any point of the hyperbola will be

If angle between asymptotes of hyperbola $\frac{{{x^2}}}{{{a^2}}} - \frac{{{y^2}}}{3} = 4$ is $\frac{\pi }{3}$, then its conjugate hyperbola is

The eccentricity of the hyperbola $\frac{{{x^2}}}{{16}} - \frac{{{y^2}}}{{25}} = 1$ is

The differential equation $\frac{{dx}}{{dy}}= \frac{{3y}}{{2x}}$ represents a family of hyperbolas (except when it represents a pair of lines) with eccentricity :

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. . . . . . . 

The value of $m$, for which the line $y = mx + \frac{{25\sqrt 3 }}{3}$, is a normal to the conic $\frac{{{x^2}}}{{16}} - \frac{{{y^2}}}{9} = 1$, is