Interference pattern and intensity variation
Interference is a phenomenon that occurs when two or more waves overlap and combine. In the case of coherent sources, the waves have the same frequency, constant phase relationship, and are emitted from two sources with a constant phase difference between them.

When these waves interfere, they create an interference pattern characterized by regions of constructive and destructive interference. The intensity of the resulting pattern depends on the superposition of the waves.

The intensity of a wave is proportional to the square of its amplitude. In an interference pattern, the total intensity at a given point is determined by the sum of the intensities of the individual waves at that point.

Amplitude of intensity variation
In this problem, it is given that the amplitude of intensity variation is 5% of the average intensity. This means that the difference between the maximum and minimum intensities in the interference pattern is 5% of the average intensity.

The amplitude of intensity variation can be calculated using the formula:
Amplitude of intensity variation = (Imax - Imin) / (Imax + Imin)

Given that the amplitude of intensity variation is 5% of the average intensity, we can write:
0.05 = (Imax - Imin) / (Imax + Imin)

Simplifying the equation, we get:
Imax - Imin = 0.05 Imax + 0.05 Imin
0.95 Imax - 1.05 Imin = 0

Relative intensities of the interfering sources
Let's assume the intensities of the two interfering sources are I1 and I2, where I1 > I2. Using the principle of superposition, the total intensity at a given point is given by:
Itotal = I1 + I2 + 2√(I1I2)cos(θ)

In the case of constructive interference, cos(θ) = 1, and in the case of destructive interference, cos(θ) = -1.

Since the amplitude of intensity variation is 5% of the average intensity, we can write:
0.05 Iaverage = (I1 + I2) / 2

Simplifying the equation, we get:
I1 + I2 = 0.1 Iaverage

Substituting this equation into the equation for total intensity, we get:
Itotal = 0.1 Iaverage + 2√(I1I2)

From the equation 0.95 Imax - 1.05 Imin = 0, we can write:
0.95 (I1 + I2) - 1.05 √(I1I2) = 0

Substituting the equation for I1 + I2, we get:
0.095 Iaverage - 1.05 √(I1I2) = 0

Simplifying the equation, we get:
√(I1I2) = 0.091 Iaverage

Squaring both sides of the equation, we get:
I1I2 = (0.091 Iaverage)^2

Since I1 > I2, we can write:
I1 = x I2

Substituting this into the equation, we get:
x I2^2 = (0.091 Iaverage)^2

Simplifying the equation