If in a hyperbola,the distance between the foci is $10$ and the transverse axis has length $8$,then the length of its latus rectum is

The equation of the hyperbola,whose eccentricity is $\sqrt{2}$ and whose foci are $16$ units apart,is

If the eccentricities of two conics $S$ and $S'$ are $e$ and $e'$ respectively,such that $e^2 + e'^2 = 3$,then both $S$ and $S'$ are:

The line $2x + \sqrt{6}y = 2$ is a tangent to the curve $x^2 - 2y^2 = 4$. The point of contact 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 $b^2$ is equal to $\frac{l}{m}(1+\sqrt{n})$,where $l$ and $m$ are co-prime numbers,then $l^2+m^2+n^2$ is equal to . . . . . . .

$A$ rectangular hyperbola passing through $(3,2)$ has its asymptotes parallel to the coordinate axes. If $(1,1)$ is the point of intersection of the two perpendicular tangents of that hyperbola,then its equation is