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

Let the product of the focal distances of the point $P(4, 2\sqrt{3})$ on the hyperbola $H: \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ be $32$. Let the length of the conjugate axis of $H$ be $p$ and the length of its latus rectum be $q$. Then $p^2 + q^2$ is equal to ......

If $\theta$ is the angle subtended by a latus rectum at the centre of the hyperbola having eccentricity $e = \frac{2}{\sqrt{7}-\sqrt{3}}$,then $\sin \theta = $

If $2x - y + 1 = 0$ is a tangent to the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{16} = 1$,then which of the following $CANNOT$ be sides of a right-angled triangle?

If the line $y = mx + c$ is a tangent to the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$,then the locus of the point $P(m, c)$ is:

What will be the equation of the chord of the hyperbola $25x^2 - 16y^2 = 400$,whose midpoint is $(5, 3)$?