The eccentricity of the hyperbola which passes through the points $(3,0)$ and $(3\sqrt{2}, 2)$ is

The equation of the tangent to the hyperbola $2x^2 - 3y^2 = 6$ which is parallel to the line $y = 3x + 4$ is:

If a directrix of a hyperbola centered at the origin and passing through the point $(4, -2 \sqrt{3})$ is $\sqrt{5}x = 4$ and $e$ is its eccentricity,then $e^2 =$

If the distance between the foci of a hyperbola $H$ is $26$ and the distance between its directrices is $\frac{50}{13}$,then the eccentricity of the conjugate hyperbola of the hyperbola $H$ is

$A$ normal to the hyperbola $4x^2 - 9y^2 = 36$ meets the coordinate axes $x$ and $y$ at $A$ and $B$,respectively. If the parallelogram $OABP$ ($O$ being the origin) is formed,then the locus of $P$ is

If $P(\theta) = (x_1, \frac{3 \sqrt{5}}{2})$,$0 < \theta < \frac{\pi}{2}$ is a point on the hyperbola $\frac{x^2}{25} - \frac{y^2}{9} = 1$,where $\theta$ is the parameter in its parametric form,then $2 x_1 + 9 \sin^2 \theta = $