The centre of the hyperbola $9x^{2} - 36x - 16y^{2} + 96y - 252 = 0$ is:

The equation of the hyperbola which passes through the point $(2,3)$ and has the asymptotes $4x+3y-7=0$ and $x-2y-1=0$ is

If a directrix of a hyperbola centered at the origin and passing through the point $(4, -2 \sqrt{3})$ is $\sqrt{5}x = 4$ and $e$ is its eccentricity,then $e^2 =$

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 ...........

The equation of the hyperbola with focus $(1, 2)$,eccentricity $e = \sqrt{3}$,and directrix $2x + y = 1$ is given by:

The locus of the point of intersection of the lines $ax \sec \theta + by \tan \theta = a$ and $ax \tan \theta + by \sec \theta = b$,where $\theta$ is the parameter,is