Let $S$ be the focus of the hyperbola $x^2-2 y^2=1$ lying on the positive $X$-axis. Let $P(-1,1)$ be a given point. Then, the area of the triangle formed by the line $P S$ with the coordinate axes is (in sq. units)

If $P\left(\frac{\pi}{6}\right)$ is a point on the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1, S, S$ are its foci and $S P+S P=2 | S P-S P$|, then $e=$

Let $e_1$ be the eccentricity of a hyperbola for which distance between its focii is 2 times the distance between its directrices and $e_2$ be the eccentricity of another hyperbola for which the length of its transverse axis is twice the length of its conjugate axis. Then, $e_1 e_2=$

61. Assertion (A) The distance between the points $p\left(\frac{\pi}{4}\right)$ and $p\left(\frac{\pi}{3}\right)$ on the hyperbola $9 x^2+16 y^2=9$ is

$$ \frac{1}{2 \sqrt{2}} \sqrt{66-33 \sqrt{2}-9 \sqrt{3}} $$

Reason (R) $x=a \cosh t, y=b \sinh t$ are the parametric equations of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$

The correct option among the following is