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

If $\theta$ is the acute angle between the asymptotes of a hyperbola $7 x^2-9 y^2=63$, then $\cos \theta=$

One of the latus recta of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ subtends an angle $2 \tan ^{-1}\left(\frac{3}{2}\right)$ at the centre of the hyperbola. If $b^2=36$ and $e$ is the eccentricity of the given hyperbola, then $\sqrt{a^2+e^2}=$

If the equation of the hyperbola having $(8,3),(0,3)$ as foci and $\frac{4}{3}$ as eccentricity is $\frac{(x-\alpha)^2}{p}-\frac{(y-\beta)^2}{q}=1$, then $p+q=$