Two coherent sources of light interfere. The intensity ratio of two...

Given here, $\dfrac{{I}_{1}}{{I}_{2}} = \dfrac{1}{4}$ or ${I}_{2} = 4{I}_{1}$

Maximum intensity is {I}_\max = {\left(\sqrt{{I}_{1}} + \sqrt{{I}_{2}}\right)}^{2} = {I}_{1} + 4{I}_{1} + 2\sqrt{{I}_{1}4{I}_{1}} = 9{I}_{1}

Minimum intensity is {I}_\min = {\left(\sqrt{{I}_{1}} - \sqrt{{I}_{2}}\right)}^{2} = {I}_{1} + 4{I}_{1} - 2\sqrt{{I}_{1}4{I}_{1}} = {I}_{1}

The ratio of \dfrac{{I}_\max + {I}_\min}{{I}_\max - {I}_\min} is

$\dfrac{9{I}_{1} + {I}_{1}}{9{I}_{1} - {I}_{1}} = \dfrac{10}{8} = \dfrac{5}{4} = \dfrac{2\alpha + 1}{\beta + 3}$

$\alpha = 2\beta = 1$

$∴ \dfrac{\alpha}{\beta} = \dfrac{2}{1} = 2$