If the distance between the foci and the distance between the directrices of the hyperbola $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$ are in the ratio $3: 2$,then $a: b$ is

The radius of the director circle of the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ is

If a hyperbola passes through the point $P(10, 16)$ and it has vertices at $(\pm 6, 0)$,then the equation of the normal to it at $P$ is

If $e_{1}$ and $e_{2}$ represent the eccentricity of the curves $16x^{2}-9y^{2}=144$ and $9x^{2}-16y^{2}=144$ respectively,then $\frac{1}{e_{1}^{2}}+\frac{1}{e_{2}^{2}}$ is equal to

If a hyperbola has length of its conjugate axis equal to $5$ and the distance between its foci is $13$,then the eccentricity of the hyperbola is

Let $A(\sec \theta, 2 \tan \theta)$ and $B(\sec \phi, 2 \tan \phi)$,where $\theta+\phi=\pi/2$,be two points on the hyperbola $2x^2-y^2=2$. If $(\alpha, \beta)$ is the point of intersection of the normals to the hyperbola at $A$ and $B$,then $(2\beta)^2$ is equal to ..... .