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 $\angle {\mathrm{SPS}}_1=\alpha$, 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 $S{P}_1$, 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

Answer: 7

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