If the line $y=m x+c$ is a common tangent to the hyperbola $\frac{x^{2}}{100}-\frac{y^{2}}{64}=1$ and the circle $x^{2}+y^{2}=36,$ then which one of the following is true?

The latus rectum of the hyperbola $9{x^2} – 16{y^2} – 18x – 32y – 151 = 0$ is

Let the latus rectum of the hyperbola $\frac{x^2}{9}-\frac{y^2}{b^2}=1$ subtend an angle of $\frac{\pi}{3}$ at the centre of the hyperbola. If $\mathrm{b}^2$ is equal to $\frac{l}{\mathrm{~m}}(1+\sqrt{\mathrm{n}})$, where $l$ and $\mathrm{m}$ are co-prime numbers, then $l^2+\mathrm{m}^2+\mathrm{n}^2$ is equal to______________.

The equation of the hyperbola whose conjugate axis is $5$ and the distance between the foci is $13$, is

Locus of foot of normal drawn from any focus to variable tangent of hyperbola $4x^2-9y^2-8x- 18y = 41$ will be