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 ..... .

$A$ hyperbola,having the transverse axis of length $2 \sin \theta$,is confocal with the ellipse $3 x^2 + 4 y^2 = 12$. Then its equation is

The locus of the midpoint of the system of parallel chords parallel to the line $y = 2x$ for the hyperbola $9x^2 - 4y^2 = 36$ is

$P(6, 3)$ is a point on the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$. If the normal at point $P$ intersects the $x$-axis at $(10, 0)$,then the eccentricity of the hyperbola is

The product of the perpendicular distances drawn from any point on the hyperbola $\frac{x^2}{9} - \frac{y^2}{4} = 1$ to its asymptotes is

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