The locus of the foot of the perpendicular drawn from any focus to a variable tangent of the hyperbola $4x^2 - 9y^2 - 8x - 18y = 41$ is:

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 $\sqrt{a^2+e^2}=$

Find the eccentricity of the conic represented by $x^{2} - y^{2} - 4x + 4y + 16 = 0$.

If the eccentricity of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ passing through the point $(4,6)$ is $2$,then the equation of the tangent to this hyperbola at $(4,6)$ is

If $\frac{x^2}{\alpha+3}+\frac{y^2}{2-\alpha}=1$ represents a hyperbola,then $\alpha$ lies in

If the lengths of the transverse and conjugate axes of a hyperbola are $8$ and $6$ respectively,find the difference of the distances of any point on the hyperbola from its foci.