Consider the hyperbola $\frac{x^{2}}{100}-\frac{y^{2}}{64}=1$ with foci at $S$ and $S_{1}$, where $S

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 $\angle S P S_{1}=\alpha$, 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

Professor PC Thomas Classes · Answer

(7) : $P N=\delta$\ \ $S_{1} P=\beta$\ We have, $S_{1} P-S P=20$\ $\Rightarrow \beta-\frac{\delta}{\sin \frac{\alpha}{2}}=20$\ $\Rightarrow \quad \beta^{2}-\frac{2 \beta \delta}{\sin \frac{\alpha}{2}}+\frac{\delta^{2}}{\sin ^{2} \frac{\alpha}{2}}=400$\ Also, $\cos \alpha=\frac{S P^{2}+\beta^{2}-656}{2 \beta \frac{\delta}{\sin \frac{\alpha}{2}}}=\frac{\frac{2 \beta \delta}{\sin \frac{\alpha}{2}}-256}{\frac{2 \beta \delta}{\sin \frac{\alpha}{2}}}$\ $\Rightarrow \frac{2 \beta \delta}{\sin \frac{\alpha}{2}}(1-\cos \alpha)=256$\ $\Rightarrow \beta \delta \cdot 2 \sin \frac{\alpha}{2}=128$\ Thus, $\frac{\beta \delta \sin \frac{\alpha}{2}}{9}=\frac{64}{9} \Rightarrow\left[\frac{\beta \delta}{9} \sin \frac{\alpha}{2}\right]=7$,\[0pt] where [.] denotes greatest integer function.\