Question

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 $\angle \operatorname{SPS}_1=\alpha$, 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 $S _1 P$ at $P _1$ Let $\delta$ be the distance of $P$ from the straight line $SP _1$, and $\beta= S _1 P$ Then the greatest integer less than or equal to $\frac{\beta \delta}{9} \sin \frac{\alpha}{2}$ is ______

Answer 1

Then the greatest integer less than or equal to $\frac{\beta \delta}{9} \sin \frac{\alpha}{2}$ is $\underline{7}$.

Answer 2

Then the greatest integer less than or equal to $\frac{\beta \delta}{9} \sin \frac{\alpha}{2}$ is $\underline{7}$.