The curve $xy = c^2$ is known as:

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

One of the latus recta of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ subtends an angle $2 \operatorname{Tan}^{-1}\left(\frac{3}{2}\right)$ at the centre of the hyperbola. If $b^2=36$ and $e$ is the eccentricity of the given hyperbola,then $\sqrt{a^2+e^2}=$

Assertion: The distance between the points $P(\frac{\pi}{4})$ and $P(\frac{\pi}{3})$ on the hyperbola $9x^2 - 16y^2 = 9$ is $\frac{1}{4} \sqrt{66 - 32\sqrt{2} - 18\sqrt{3}}$.
Reason: $x = a \cosh t, y = b \sinh t$ are the parametric equations of the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$.

If $e_{1}$ and $e_{2}$ are the eccentricities of a hyperbola $3x^{2} - 3y^{2} = 25$ and its conjugate,then

The equation of the normal to the curve $x=a \cosh(t), y=b \sinh(t)$ at any point $t$ is