Subject: Mathematics — Hyperbola
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
$$\angle SPS_{1} = \alpha,\quad\text{with }\alpha < \frac{\pi}{2}.$$
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: 7
##
Solution: [IMAGE] In $\bigtriangleup S_{1}QP$,
$$\sin\frac{\alpha}{2} = \frac{S_{1}Q}{\beta} \Rightarrow S_{1}Q = \beta\sin\frac{\alpha}{2}$$
Product of distances of any tangent from two foci is $b^{2}$.
$$\delta \cdot S_{1}Q = \delta \cdot \beta\sin\frac{\alpha}{2} = b^{2}$$
So,
$$\frac{\beta\delta\sin\frac{\alpha}{2}}{9} = \frac{b^{2}}{9} = \frac{64}{9}$$
$$\left\lbrack \frac{64}{9} \right\rbrack = 7$$