If $P(\frac{\pi}{4})$ and $Q(\frac{3 \pi}{4})$ are two points on the hyperbola $4 x^2-y^2-8 x-2 y-13=0$ in parametric form,then the distance between $P$ and $Q$ is

The tangent at an extremity (in the first quadrant) of the latus rectum of the hyperbola $\frac{x^2}{4} - \frac{y^2}{5} = 1$ meets the $x$-axis and $y$-axis at $A$ and $B$ respectively. Then $(OA)^2 - (OB)^2$,where $O$ is the origin,equals

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 of a hyperbola through one focus subtends an angle of $60^{\circ}$ at the other focus,then its eccentricity is

The descending order of magnitude of the eccentricities of the following hyperbolas is:
$A$. $A$ hyperbola whose distance between foci is three times the distance between its directrices.
$B$. Hyperbola in which the transverse axis is twice the conjugate axis.
$C$. Hyperbola with asymptotes $x+y+1=0$ and $x-y+3=0$.

Equation of one of the tangents passing through $(2,8)$ to the hyperbola $5 x^2-y^2=5$ is