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 = $

The product of the perpendicular distances from any point on the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ to its asymptotes is

Let $L(ae, b^2/a)$ be the end of the latus rectum of the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ lying in the first quadrant,and let $S(ae, 0)$ be the focus of the given hyperbola. Given $L$ is $(x_1, 4)$ and $S$ is $(8, y_1)$,find the length of its transverse axis.

Let tangents drawn from point $C(0,-b)$ to the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ touch the hyperbola at points $A$ and $B$. If $\Delta ABC$ is a right-angled triangle,then $\frac{a^2}{b^2}$ is equal to -

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

The locus of the point of intersection of lines $(x + y)t = a$ and $x - y = at$,where $t$ is the parameter,is