Equation of the hyperbola with eccentricity $\frac{3}{2}$ and foci at $\left(\pm 2,0\right)$ is $5x^2-4y^2=k^2$. What is the value of $k$ $?$

The hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ passes through the point $\left(3\sqrt{5},1\right)$ and the length of its latus rectum is $\frac{4}{3}$ units. The length of the conjugate axis is

If the foci of the ellipse $\frac{x^2}{16}+\frac{y^2}{b^2}=1$ and the hyperbola $\frac{x^2}{144}-\frac{y^2}{81}=\frac{1}{25}$ coincide, then the value of $b^2$ is