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 $\_\_\_\_$ .
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 $\_\_\_\_$ .