With one focus of the hyperbola $\frac{{{x^2}}}{9}\,\, - \,\,\frac{{{y^2}}}{{16}}\,\, = \,\,1$ as the centre , a circle is drawn which is tangent to the hyperbola with no part of the circle being outside the hyperbola. The radius of the circle is

Consider a branch of the hyperbola $x^2-2 y^2-2 \sqrt{2} x-4 \sqrt{2} y-6=0$ with vertex at the point $A$. Let $B$ be one of the end points of its latus rectum. If $\mathrm{C}$ is the focus of the hyperbola nearest to the point $\mathrm{A}$, then the area of the triangle $\mathrm{ABC}$ is

A hyperbola whose transverse axis is along the major axis of then conic, $\frac{{{x^2}}}{3} + \frac{{{y^2}}}{4} = 4$ and has vertices at the foci of this conic . If the eccentricity of the hyperbola is $\frac{3}{2}$ , then which of the following points does $NOT$ lie on it ?

Let the eccentricity of the hyperbola $H : \frac{ x ^{2}}{ a ^{2}}-\frac{ y ^{2}}{ b ^{2}}=1$ be $\sqrt{\frac{5}{2}}$ and length of its latus rectum be $6 \sqrt{2}$, If $y =2 x + c$ is a tangent to the hyperbola $H$, then the value of $c ^{2}$ is equal to

The equation of the hyperbola referred to the axis as axes of co-ordinate and whose distance between the foci is $16$ and eccentricity is $\sqrt 2 $, is

Let the hyperbola $H : \frac{ x ^{2}}{ a ^{2}}- y ^{2}=1$ and the ellipse $E: 3 x^{2}+4 y^{2}=12$ be such that the length of latus rectum of $H$ is equal to the length of latus rectum of $E$. If $e_{ H }$ and $e_{ E }$ are the eccentricities of $H$ and $E$ respectively, then the value of $12\left( e _{ H }^{2}+ e _{ E }^{2}\right)$ is equal to