Find the equation of the hyperbola satisfying the given conditions: Foci $(0, \pm 13)$,the conjugate axis is of length $24$.

Let the lengths of the transverse and conjugate axes of a hyperbola in standard form be $2a$ and $2b$,respectively,and one focus and the corresponding directrix of this hyperbola be $(-5, 0)$ and $5x + 9 = 0$,respectively. If the product of the focal distances of a point $(\alpha, 2\sqrt{5})$ on the hyperbola is $p$,then $4p$ is equal to . . . . . . .

Find the equation of the hyperbola satisfying the given conditions: Vertices $(0, \pm 5)$,foci $(0, \pm 8)$.

If the point $(1,1)$ and the origin lie in the same region with respect to the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{1} = 1$ $(a > 0)$,then the range of $a$ is

Consider a hyperbola $H$ having its centre at the origin and foci on the $x$-axis. Let $C_1$ be a circle touching the hyperbola $H$ and having its centre at the origin. Let $C_2$ be a circle touching the hyperbola $H$ at its vertex and having its centre at one of its foci. If the areas (in sq. units) of $C_1$ and $C_2$ are $36 \pi$ and $4 \pi$,respectively,then the length (in units) of the latus rectum of $H$ is

The locus of the point of intersection of the lines $\sqrt{3}x - y - 4\sqrt{3}k = 0$ and $\sqrt{3}kx + ky - 4\sqrt{3} = 0$ for different values of $k$ is