If the eccentricity of the hyperbola $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$ is $\frac{5}{4}$ and $2x+3y-6=0$ is a focal chord of the hyperbola,then the length of the transverse axis is equal to

$A$ 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

The equation of the hyperbola whose directrix is $x + 2y = 1$,focus $(2, 1)$ and eccentricity $e = 2$ is:

Find the length of the latus rectum of the hyperbola $16x^2 - 9y^2 = 144$. (in $/3$)

Find the equation of the hyperbola whose foci are $(-2, 0)$ and $(2, 0)$ and eccentricity is $2$.

The product of the perpendicular distances from any point on the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ to its asymptotes is