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 -

If the two tangents drawn on hyperbola $\frac{{{x^2}}}{{{a^2}}} - \frac{{{y^2}}}{{{b^2}}} = 1$ in such a way that the product of their gradients is ${c^2}$, then they intersects on the curve

Find the equation of the hyperbola satisfying the give conditions: Foci $(0,\,\pm 13),$ the conjugate axis is of length $24.$

If the tangent and normal to a rectangular hyperbola $xy = c^2$ at a variable point cut off intercept  $a_1, a_2$ on $x-$ axis and $b_1, b_2$ on $y-$ axis, then $(a_1a_2 + b_1b_2)$ is

The foci of the hyperbola $9{x^2} - 16{y^2} = 144$ are

For hyperbola $\frac{{{x^2}}}{{{{\cos }^2}\alpha }} - \frac{{{y^2}}}{{{{\sin }^2}\alpha }} = 1$ which of the following remains constant with change in $'\alpha '$