Find the equation of the hyperbola satisfying the given conditions: Foci $(\pm 5, 0)$,the transverse axis is of length $8$.

If $(1,2)$ is the focus,$x+2y=0$ is the directrix,and $\sqrt{2}$ is the eccentricity of a hyperbola,then the equation of the hyperbola is

If the lengths of the transverse axis and the latus rectum of a hyperbola are $6$ and $\frac{8}{3}$ respectively,then the equation of the hyperbola is $ . . . . . . $

If the line $y = mx + 7\sqrt{3}$ is normal to the hyperbola $\frac{x^2}{24} - \frac{y^2}{18} = 1$,then a value of $m$ 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 $............$.

Statement $ (A) $: The point $(5, -4)$ lies inside the hyperbola $y^2 - 9x^2 + 1 = 0$.
Reason $(R)$: $A$ point $(x_1, y_1)$ lies inside the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ if $\frac{x_1^2}{a^2} - \frac{y_1^2}{b^2} - 1 < 0$.