The length of the latus rectum of the hyperbola $9x^2 - 16y^2 + 72x - 32y - 16 = 0$ is

If $e_{1}$ and $e_{2}$ represent the eccentricity of the curves $16x^{2}-9y^{2}=144$ and $9x^{2}-16y^{2}=144$ respectively,then $\frac{1}{e_{1}^{2}}+\frac{1}{e_{2}^{2}}$ is equal to

If $\frac{x^{2}}{36}-\frac{y^{2}}{k^{2}}=1$ is a hyperbola,then which of the following statements can be true?

The combined equation of the asymptotes of the hyperbola $2x^2 + 5xy + 2y^2 + 4x + 5y = 0$ is:

Let $X$-axis be the transverse axis and $Y$-axis be the conjugate axis of a hyperbola $H$. Let the eccentricity of $H$ be the reciprocal of the eccentricity of the ellipse $\frac{x^2}{4} + \frac{y^2}{2} = 1$. If $(5, 4)$ is a point on $H$,then the length of the transverse axis of $H$ is

$A$ hyperbola passes through the foci of the ellipse $\frac{x^{2}}{25}+\frac{y^{2}}{16}=1$ and its transverse and conjugate axes coincide with the major and minor axes of the ellipse,respectively. If the product of their eccentricities is $1$,then the equation of the hyperbola is ...... .