If $e_1$ and $e_2$ are the eccentricities of the hyperbola $16 x^2 - 9 y^2 = 1$ and its conjugate respectively,then $3 e_1 = $

$A$ hyperbola having its centre at the origin passes through the point $(5, 2)$ and has a transverse axis of length $8$ along the $X$-axis. What is the eccentricity of its conjugate hyperbola?

Let $H_1: \frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ and $H_2:-\frac{x^2}{A^2}+\frac{y^2}{B^2}=1$ be two hyperbolas having lengths of latus rectums $15 \sqrt{2}$ and $12 \sqrt{5}$ respectively. Let their eccentricities be $e_1=\sqrt{\frac{5}{2}}$ and $e_2$ respectively. If the product of the lengths of their transverse axes is $100 \sqrt{10}$,then $25 e_2^2$ is equal to . . . . . . .

If the equation of one asymptote of the hyperbola $14 x^2+38 x y+20 y^2+x-7 y-91=0$ is $7 x+5 y-3=0$,then the other asymptote is

For a hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$,if the length of the transverse axis is $8$ and the distance between the foci is $2\sqrt{41}$,then the length of its latus rectum is

Let $(1, 2)$ be the focus and $x+y+1=0$ be the directrix of a hyperbola $H$. If $\sqrt{3}$ is the eccentricity of $H$,then its equation is