Two light beams of intensities in the ratio of 9: 4 are allowed to...

Data provided in the question is, the ratio of the intensities,

$\dfrac{{I}_{1}}{{I}_{2}} = \dfrac{9}{4}$

After taking square root of above equation on both sides, we get,

$\dfrac{\sqrt{{I}_{1}}}{\sqrt{{I}_{2}}} = \dfrac{3}{2}$

Apply componendo and dividendo,
$⇒ \dfrac{\sqrt{{I}_{1}} + \sqrt{{I}_{2}}}{\sqrt{{I}_{1}} - \sqrt{{I}_{2}}} = \dfrac{3 + 2}{3 - 2} = \dfrac{5}{1}$
⇒ {\left(\dfrac{\sqrt{{I}_{1}} + \sqrt{{I}_{2}}}{\sqrt{{I}_{1}} - \sqrt{{I}_{2}}}\right)}^{2} = \dfrac{{I}_\max}{{I}_\min} = \dfrac{25}{1}