If the line $y=mx+c$ 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?

The equation of a hyperbola whose asymptotes are $3x \pm 5y = 0$ and vertices are $(\pm 5, 0)$ is

When is the line $ℓx + my + n = 0$ a tangent to the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$?

Let $H: \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ be a hyperbola such that the distance between its foci is $6$ and the distance between its directrices is $\frac{8}{3}$. If the line $x = k$ intersects the hyperbola $H$ at the points $A$ and $B$ such that the area of the triangle $AOB$ is $4\sqrt{15}$, where $O$ is the origin, then $a^2$ equals

The line $y = mx + c$ touches the curve $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$,if

If $\sqrt{5} y - \sqrt{8} = 0$ is the equation of the directrix of a hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} + 1 = 0$ and $\frac{\sqrt{5}}{2}$ is its eccentricity,then $\frac{1}{a} =$