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 locus of the point of intersection of any two perpendicular tangents to the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ is a circle,which is called the director circle of the hyperbola. What is the equation of this circle?

If $L_1=0$ and $L_2=0$ are the asymptotes of the hyperbola $9x^2-4y^2+36x+8y-4=0$,then the product of the perpendicular distances from the point $(1,1)$ to the lines $L_1=0$ and $L_2=0$ is

The asymptotes of the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ form with any tangent to the hyperbola a triangle whose area is $a^2 \tan \lambda$ in magnitude. Then its eccentricity $e$ is:

If the length of the transverse and conjugate axes of a hyperbola are $8$ and $6$ respectively,then the difference of the focal distances of any point on the hyperbola will be

When is the line $ℓx + my + n = 0$ a tangent to the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$?