The interference pattern is obtained with two coherent light sources of intensity ratio $9: 1$. The ratio of $\dfrac{I_{\text {MAX }}+I_{\text {MIN }}}{I_{\text {MAX }}-I_{\text {MIN }}}$ is $\dfrac{\alpha}{\beta}$. The values of $\alpha$ and $\beta$ are:

Let the intensities of the two coherent sources be $I_1$ and $I_2$. Given the ratio $I_1 : I_2 = 9 : 1$, we can write $I_1 = 9k$ and $I_2 = k$ for some constant $k$.

The amplitudes $a_1$ and $a_2$ are proportional to the square root of the intensities, so $a_1 = \sqrt{9k} = 3\sqrt{k}$ and $a_2 = \sqrt{k}$.

The maximum intensity $I_{\text{MAX}}$ and minimum intensity $I_{\text{MIN}}$ in an interference pattern are given by $I_{\text{MAX}} = (\sqrt{I_1} + \sqrt{I_2})^2$ and $I_{\text{MIN}} = (\sqrt{I_1} - \sqrt{I_2})^2$.

Substituting the values, $I_{\text{MAX}} = (3\sqrt{k} + \sqrt{k})^2 = (4\sqrt{k})^2 = 16k$ and $I_{\text{MIN}} = (3\sqrt{k} - \sqrt{k})^2 = (2\sqrt{k})^2 = 4k$.

The required ratio is $\dfrac{I_{\text{MAX}} + I_{\text{MIN}}}{I_{\text{MAX}} - I_{\text{MIN}}} = \dfrac{16k + 4k}{16k - 4k} = \dfrac{20k}{12k} = \dfrac{20}{12} = \dfrac{5}{3}$.

Comparing this with $\dfrac{\alpha}{\beta}$, we get $\alpha = 5$ and $\beta = 3$.

Answer: 5 and 3