Find the equations of the hyperbola satisfying the given conditions:

foci $(\pm 3\sqrt{5}, 0)$, the latus-rectum $=8$

Find the equation of the hyperbola satisfying the given conditions:

foci $(\pm 4, 0)$, the latus-rectum $=12$

Find the equation of the hyperbola satisfying the given conditions:

vertices $(0, \pm 6), e=\frac{5}{3}$

Find the equation of the hyperbola satisfying the given conditions:

foci $(0, \pm \sqrt{10})$, passing through $(2, 3)$

Find the equations of the hyperbola satisfying the given conditions:

foci $(0, \pm 12)$, latus-rectum $=36$

Write the eccentricity of the hyperbola $9x^2-16y^2=144$.

The eccentricity of the hyperbola whose latus-rectum is half of its transverse axis is $\sqrt{\frac{k}{2}}$

Find the value of $k$