If the line $y=m x+c$ is a common tangent to the hyperbola $\frac{x^{2}}{100}-\frac{y^{2}}{64}=1$ and the circle $x^{2}+y^{2}=36,$ then which one of the following is true?

Let tangents drawn from point $C(0,-b)$ to hyperbola $\frac{{{x^2}}}{{{a^2}}} - \frac{{{y^2}}}{{{b^2}}} = 1$ touches hyperbola at points $A$ and $B.$ If $\Delta ABC$ is a right angled triangle, then $\frac{a^2}{b^2}$ is equal to -

Find the equation of the hyperbola satisfying the give conditions : Vertices $(\pm 7,\,0)$,  $e=\frac{4}{3}$

An ellipse $E: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ passes through the vertices of the hyperbola $H: \frac{x^{2}}{49}-\frac{y^{2}}{64}=-1$. Let the major and minor axes of the ellipse $E$ coincide with the transverse and conjugate axes of the hyperbola $H$. Let the product of the eccentricities of $E$ and $H$ be $\frac{1}{2}$. If $l$ is the length of the latus rectum of the ellipse $E$, then the value of $113\,l$ is equal to $....$

If $(a -2)x^2 + ay^2 = 4$ represents rectangular hyperbola, then $a$ equals :-

The equation of the hyperbola whose foci are the foci of the ellipse $\frac{{{x^2}}}{{25}} + \frac{{{y^2}}}{9} = 1$ and the eccentricity is $2$, is