The tangent to the hyperbola $xy = c^2$  at the point $P$  intersects the $x-$ axis at $T$ and the $y-$ axis at $T'$. The normal to the hyperbola at $P$ intersects the $ x-$ axis at $N$  and the $y-$ axis at $N'$. The areas of the triangles $PNT$  and $PN'T' $ are $ \Delta$  and $ \Delta ' $ respectively, then $\frac{1}{\Delta }\,\, + \,\,\frac{1}{{\Delta '}}\,$ is

Eccentricity of the curve ${x^2} - {y^2} = {a^2}$ 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 $\angle SPS _1=\alpha$, 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 $SP _1$, and $\beta= S _1 P$. Then the greatest integer less than or equal to $\frac{\beta \delta}{9} \sin \frac{\alpha}{2}$ is. . . . . . . 

For hyperbola $\frac{{{x^2}}}{{{{\cos }^2}\alpha }} - \frac{{{y^2}}}{{{{\sin }^2}\alpha }} = 1$ which of the following remain constant if $\alpha$ varies

$C$ the centre of the hyperbola $\frac{{{x^2}}}{{{a^2}}} - \frac{{{y^2}}}{{{b^2}}} = 1$. The tangents at any point $P$ on this hyperbola meets the straight lines $bx - ay = 0$ and $bx + ay = 0$ in the points $Q$ and $R$ respectively. Then $CQ\;.\;CR = $