If the eccentricity of a hyperbola $\frac{x^2}{9} - \frac{y^2}{b^2} = 1,$ which passes through $(K, 2),$ is $\frac{\sqrt{13}}{3},$ then the value of $K^2$ is

The eccentricity of the conjugate hyperbola of the hyperbola ${x^2} - 3{y^2} = 1$ is

Let $P(a \sec \theta, b \tan \theta)$ and $Q(a \sec \phi, b \tan \phi)$ be two points such that $\theta+\phi=\frac{\pi}{2}$ on the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$. If $(h, k)$ is the point of intersection of the normals at $P$ and $Q$,then $k=$

$A$ common tangent to $9x^{2}-16y^{2}=144$ and $x^{2}+y^{2}=9$ is

The lines of the form $x \cos \phi + y \sin \phi = P$ are chords of the hyperbola $4x^2 - y^2 = 4a^2$ which subtend a right angle at the centre of the hyperbola. If these chords touch a circle with centre at $(0,0)$,then the radius of that circle is

The angle between the asymptotes of the hyperbola $x^2-3y^2=3$ is