The angle between the tangents drawn from the point $(2\sqrt{2}, 1)$ to the hyperbola $16x^2 - 25y^2 = 400$ is ........

If the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ has eccentricity $e = \frac{5}{4}$ and the length of the latus rectum equal to $9$,then $ab$ is equal to

The equation $3x^2 + 7xy + 2y^2 + 5x + 5y + 2 = 0$ represents:

The equation of the normal to the curve $3x^2 - y^2 = 8$,which is parallel to the line $x + 3y = 10$,is

The product of the lengths of the perpendiculars from any point on the hyperbola $x^2 - y^2 = 8$ to its asymptotes 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: