If the common tangents of x^ 2 + y^ 2 = r^ 2 and x^ 2 16 + y^ 2 9 =...

Let, the equation of tangent in slope form for the circle be $y = mx \pm r\sqrt{1 + m^{2}}$

and the equation of tangent in slope form for the ellipse be $y = mx \pm \sqrt{16m^{2} + 9}$

Now, for common tangets,
$y = mx \pm r\sqrt{1 + m^{2}}$ is same as $y = mx \pm \sqrt{16m^{2} + 9}$
$⇒ r^{2} + r^{2}m^{2} = 16m^{2} + 9 ⇒ \left(r^{2} - 16\right)m^{2} + \left(r^{2} - 9\right) = 0$

Since, here ${m}_{1}{m}_{2} = - 1$
$⇒ 2r^{2} = 25 ⇒ r = \dfrac{5}{2}$

So, the side of the square $= 2r = 5\sqrt{2}$
$⇒$ area of the square is $50$& sq. units