If for a hyperbola the ratio of length of conjugate Axis to the length of transverse  axis is $3 : 2$ then the ratio of distance between the focii to the distance between the two directrices is

The product of the lengths of perpendiculars drawn from any point on the hyperbola $x^2 -2y^2 -2=0$  to its asymptotes is 

Find the coordinates of the foci and the vertices, the eccentricity, and the length of the latus rectum of the hyperbola $9 y^{2}-4 x^{2}=36$

Equation of hyperbola with asymptotes $3x - 4y + 7 = 0$ and $4x + 3y + 1 = 0$ and which passes through origin is

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

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