If $e_1$ is the eccentricity of the hyperbola $x = \sec \theta, y = \sqrt{2} \tan \theta$ and $e_2$ is the eccentricity of the hyperbola $x = \sqrt{2} \sec \theta, y = \tan \theta$,then $\frac{e_2^2}{e_1^2} = $

If the latus rectum of a hyperbola subtends an angle of $120^{\circ}$ at its centre,then its eccentricity is

The equation of the asymptotes of the hyperbola $2 x^2+5 x y+2 y^2-11 x-7 y-4=0$ is

Find the condition for the curves $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ and $xy = c^2$ to intersect orthogonally.

$A$ hyperbola passes through a focus of the ellipse $\frac{x^2}{169}+\frac{y^2}{25}=1$. Its transverse and conjugate axes coincide respectively with the major and minor axes of the ellipse. The product of their eccentricities is $1$. Then,the equation of the hyperbola is

Let $H_{n} = \frac{x^2}{1+n} - \frac{y^2}{3+n} = 1$,where $n \in N$. Let $k$ be the smallest even value of $n$ such that the eccentricity of $H_{k}$ is a rational number. If $l$ is the length of the latus rectum of $H_{k}$,then $21l$ is equal to $.......$.