The equation of the common tangent to the curves $y^2 = 8x$ and $xy = -1$ is

Find the coordinates of the foci and the vertices, the eccentricity,the length of the latus rectum of the hyperbolas : $y^{2}-16 x^{2}=16$ 

If the centre, vertex and focus of a hyperbola be $(0, 0), (4, 0)$ and $(6, 0)$ respectively, then the equation of the hyperbola is

Latus rectum of the conic satisfying the differential equation, $ x dy + y dx = 0$  and passing through the point $ (2, 8) $ is :

Find the coordinates of the foci and the vertices, the eccentricity, and the length of the latus rectum of the hyperbola $5 y^{2}-9 x^{2}=36$

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______________.