What kind of hyperbola does the equation $9x^2 - 16y^2 - 18x + 32y - 151 = 0$ represent?

If $e_1$ is the eccentricity of the hyperbola $x = \sec \theta, y = \sqrt{2} \tan \theta$ and $e_2$ is the eccentricity of the hyperbola $x = \sqrt{2} \sec \theta, y = \tan \theta$,then $\frac{e_2^2}{e_1^2} = $

The descending order of magnitude of the eccentricities of the following hyperbolas is:
$A$. $A$ hyperbola whose distance between foci is three times the distance between its directrices.
$B$. Hyperbola in which the transverse axis is twice the conjugate axis.
$C$. Hyperbola with asymptotes $x+y+1=0$ and $x-y+3=0$.

If $\theta$ is the acute angle between the tangents drawn from the point $(1,1)$ to the hyperbola $4x^2 - 5y^2 = 20$,then $\tan \theta = $

For the variable $t,$ the locus of the points of intersection of lines $x-2y=t$ and $x+2y=\frac{1}{t}$ is

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