If $l_1$ and $l_2$ are the lengths of the perpendiculars drawn from a point on the hyperbola $5x^2 - 4y^2 - 20 = 0$ to its asymptotes,then $\frac{l_1^2 l_2^2}{100} = $

For the hyperbola $x^2 \sec^2 \theta - y^2 \csc^2 \theta = 1$,which of the following remains constant when $\theta$ varies?

The locus of the midpoints of the chords of the circle $x^2 + y^2 = a^2$ which touch the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ is

Consider the hyperbola $\frac{x^2}{100}-\frac{y^2}{64}=1$ with foci at $S$ and $S_1$,where $S$ lies on the positive $x$-axis. Let $P$ be a point on the hyperbola,in the first quadrant. Let $\angle SPS_1 = \alpha$,with $\alpha < \frac{\pi}{2}$. The straight line passing through the point $S$ and having the same slope as that of the tangent at $P$ to the hyperbola,intersects the straight line $S_1P$ at $P_1$. Let $\delta$ be the distance of $P$ from the straight line $SP_1$,and $\beta = S_1P$. Then the greatest integer less than or equal to $\frac{\beta \delta}{9} \sin \frac{\alpha}{2}$ is:

The tangent at an extremity (in the first quadrant) of the latus rectum of the hyperbola $\frac{x^2}{4} - \frac{y^2}{5} = 1$ meets the $x$-axis and $y$-axis at $A$ and $B$ respectively. Then $(OA)^2 - (OB)^2$,where $O$ is the origin,equals

If a circle intersects the hyperbola $xy = 1$ at four points $(x_r, y_r)$ for $r = 1, 2, 3, 4$,then: