Two coherent sources with different intensities emit waves that...

Let the intensities of the two coherent sources be $I_1$ and $I_2$.

The maximum intensity in the interference pattern is given by I_\max = (\sqrt{I_1} + \sqrt{I_2})^2

The minimum intensity is given by I_\min = (\sqrt{I_1} - \sqrt{I_2})^2

Given that the ratio of maximum to minimum intensity is $25$:

\dfrac{I_\max}{I_\min} = \dfrac{(\sqrt{I_1} + \sqrt{I_2})^2}{(\sqrt{I_1} - \sqrt{I_2})^2} = 25

Taking the square root on both sides:

$\dfrac{\sqrt{I_1} + \sqrt{I_2}}{\sqrt{I_1} - \sqrt{I_2}} = 5$

$\sqrt{I_1} + \sqrt{I_2} = 5\sqrt{I_1} - 5\sqrt{I_2}$

$6\sqrt{I_2} = 4\sqrt{I_1}$

$\dfrac{\sqrt{I_1}}{\sqrt{I_2}} = \dfrac{6}{4} = \dfrac{3}{2}$

Squaring both sides, the ratio of the intensities is:

$\dfrac{I_1}{I_2} = \dfrac{9}{4}$