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

$ S P- PP =20$
$ \beta-\frac{\delta}{\sin \frac{\alpha}{2}}=20$
$ \beta^2+\frac{\delta^2}{\sin ^2 \frac{\alpha}{2}}-400=\frac{2 \beta \delta}{\sin \frac{\alpha}{2}} $
$ \frac{1}{ SP }=\frac{\sin \frac{\alpha}{2}}{\delta} $
$\cos \alpha=\frac{ SP ^2+\beta^2-656}{2 \beta \frac{\delta}{\sin \frac{\alpha}{2}}}$
$ =\frac{\frac{2 \beta \delta}{\sin \frac{\alpha}{2}}-256}{\frac{2 \beta S}{\sin \frac{\alpha}{2}}}=\cos \alpha$
$ \frac{\lambda-128}{\lambda}=\cos \alpha$
$ \lambda(1-\cos \alpha)=128$
$\frac{\beta \delta}{\sin \frac{\alpha}{2}} \cdot 2 \sin ^2 \frac{\alpha}{2}=128 $
$ \frac{\beta \delta}{9} \sin \frac{\alpha}{2}=\frac{64}{9} $ $\Rightarrow\left[\frac{\beta \delta}{9} \sin \frac{\alpha}{2}\right]=7 $ where [.] denotes greatest integer function