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, write the value of $b^2.$

Distance between the directrices of the hyperbola is $k\sqrt{2}$ where  $x = 8 \sec \theta$, $y = 8$ $\tan$ $\theta .$

Then find the value of $k$.

Write the equation of the hyperbola whose vertices are $(\pm 3, 0)$ and foci at $(\pm 5, 0).$