To find the ratio of the intensities of the maximum to the minimum in an interference pattern formed by two monochromatic light beams with intensity ratios of 1:9, you use the formula:

$ \frac{I_{\max }}{I_{\min }} = \frac{\left(\sqrt{I_1} + \sqrt{I_2}\right)^2}{\left(\sqrt{I_1} - \sqrt{I_2}\right)^2} $

Given that the intensity ratio is 1:9, we have $I_1 = 1$ and $I_2 = 9$.

Calculate $\sqrt{I_1}$ and $\sqrt{I_2}$:

$ \sqrt{I_1} = \sqrt{1} = 1 $

$ \sqrt{I_2} = \sqrt{9} = 3 $

Substitute these values into the formula:

$ \frac{I_{\max }}{I_{\min }} = \frac{(1 + 3)^2}{(1 - 3)^2} = \frac{4^2}{2^2} = \frac{16}{4} = 4 $

Thus, the ratio of the intensities of the maximum to the minimum is 4:1.