If $P(x_1, y_1), Q(x_2, y_2), R(x_3, y_3)$ and $S(x_4, y_4)$ are $4$ concyclic points on the rectangular hyperbola $xy = c^2$,the coordinates of the orthocentre of the triangle $PQR$ are:

The equation of the director circle of the hyperbola $\frac{x^2}{16} - \frac{y^2}{4} = 1$ is given by

If the latus rectum subtends a right angle at the center of the hyperbola,then its eccentricity is

The locus of the point of intersection of the lines $ax \sec \theta + by \tan \theta = a$ and $ax \tan \theta + by \sec \theta = b$,where $\theta$ is the parameter,is

Let the foci of the hyperbola coincide with the foci of the ellipse $\frac{x^{2}}{36}+\frac{y^{2}}{16}=1$. If the eccentricity of the hyperbola is $5$, then the length of its latus rectum is:

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 $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$.