The product of the lengths of the perpendiculars from any point on the hyperbola $x^2-y^2=16$ to its asymptotes is

The foci of the hyperbola $9x^2 - 16y^2 = 144$ are

Let $H : \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$,$a > 0, b > 0$,be a hyperbola such that the sum of lengths of the transverse and the conjugate axes is $4(2\sqrt{2}+\sqrt{14})$. If the eccentricity of $H$ is $\frac{\sqrt{11}}{2}$,then the value of $a^{2}+b^{2}$ is equal to

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

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

The tangents drawn to the hyperbola $5x^2 - 9y^2 = 90$ through a variable point $P$ make the angles $\alpha$ and $\beta$ with its transverse axis. If $\alpha$ and $\beta$ are complementary angles,then the locus of $P$ is