The values of $m$ for which the line $y=mx+2$ is a tangent to the hyperbola $4x^2-9y^2=36$ are

The locus of the point of intersection of the tangents drawn at the extremities of a normal chord of the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ is

If the angle between the asymptotes of a hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ is $2 \tan^{-1}\left(\frac{b}{a}\right) = 2 \tan^{-1}\left(\frac{2}{3}\right)$ and $a^2-b^2=45$,then $ab=$

Find the equation of the hyperbola satisfying the given conditions: Foci $(\pm 3 \sqrt{5}, 0)$,the latus rectum is of length $8$.

The distance between the tangents to the hyperbola $\frac{x^2}{20} - \frac{3y^2}{4} = 1$ which are parallel to the line $x + 3y = 7$ is

The equation of the hyperbola having its eccentricity $e = 2$ and the distance between its foci as $8$ is: