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 ________.

The line $$2x + y = 1$$ is tangent to the hyperbola $${{{x^2}} \over {{a^2}}} - {{{y^2}} \over {{b^2}}} = 1$$.

If this line passes through the point of intersection of the nearest directrix and the $$x$$-axis, then the eccentricity of the hyperbola is