Two coherent light sources having intensity in the ratio $$2x$$ produce an interference pattern. The ratio $$\frac{I_{max} - I_{min}}{I_{max} + I_{min}}$$ will be:

Let the intensities of the two coherent light sources be $$I_1 = 2x$$ and $$I_2 = 1$$ (ratio $$2x : 1$$).

For an interference pattern, the maximum and minimum intensities are:

$$I_{max} = \left(\sqrt{I_1} + \sqrt{I_2}\right)^2 = \left(\sqrt{2x} + 1\right)^2$$

$$I_{min} = \left(\sqrt{I_1} - \sqrt{I_2}\right)^2 = \left(\sqrt{2x} - 1\right)^2$$

Computing $$I_{max} - I_{min}$$:

$$I_{max} - I_{min} = \left(\sqrt{2x} + 1\right)^2 - \left(\sqrt{2x} - 1\right)^2 = 4\sqrt{2x}$$

Computing $$I_{max} + I_{min}$$:

$$I_{max} + I_{min} = \left(\sqrt{2x} + 1\right)^2 + \left(\sqrt{2x} - 1\right)^2 = 2(2x + 1)$$

Therefore the required ratio is:

$$\frac{I_{max} - I_{min}}{I_{max} + I_{min}} = \frac{4\sqrt{2x}}{2(2x + 1)} = \frac{2\sqrt{2x}}{2x + 1}$$

The correct answer is Option (3): $$\frac{2\sqrt{2x}}{2x + 1}$$.