The length of the latus rectum of the curve $xy = 7x + 5y$ is

If the line $ax + by = 1$ is normal to the hyperbola $\frac{x^2}{p^2} - \frac{y^2}{q^2} = 1$,then $\frac{p^2}{a^2} - \frac{q^2}{b^2}$ is equal to (where $a, b, p, q \in R^+$):

The locus of the midpoints of the chord of the circle $x^{2}+y^{2}=25$ which is tangent to the hyperbola $\frac{x^{2}}{9}-\frac{y^{2}}{16}=1$ is

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

The distance between the tangents drawn to the hyperbola $3x^2 - y^2 = 3$,which are parallel to the line $y = 2x + 4$,is:

The length of the latus rectum and directrices of a hyperbola with eccentricity $e$ are $9$ and $x = \pm \frac{4}{\sqrt{13}}$,respectively. Let the line $y - \sqrt{3}x + \sqrt{3} = 0$ touch this hyperbola at $(x_0, y_0)$. If $m$ is the product of the focal distances of the point $(x_0, y_0)$,then $4e^2 + m$ is equal to ...........