Ratio of amplitudes of two coherent sources producing an interference pattern is given as 3:4. We need to find the ratio of intensities at maxima and minima.

Explanation:
Interference is the phenomenon that occurs when two or more waves combine to form a resultant wave. It can be constructive or destructive depending on the phase difference between the waves.

When two coherent sources produce an interference pattern, the resultant intensity at a particular point is given by the square of the algebraic sum of the amplitudes of the two waves.

Let the amplitudes of the two coherent sources be A1 and A2, and their ratio is given as 3:4. So, we can assume A1 = 3x and A2 = 4x, where x is a constant.

The intensity at a point due to the first source is directly proportional to the square of its amplitude, i.e., I1 ∝ |A1|^2 = (3x)^2 = 9x^2.

Similarly, the intensity at the same point due to the second source is directly proportional to the square of its amplitude, i.e., I2 ∝ |A2|^2 = (4x)^2 = 16x^2.

Now, let's consider a point where the interference is constructive, resulting in a maximum intensity. The two waves will have the same phase at this point.

The resultant amplitude at this point will be the sum of the amplitudes of the two waves: A = A1 + A2 = 3x + 4x = 7x.

The resultant intensity at this point will be the square of the resultant amplitude: I_max ∝ |A|^2 = (7x)^2 = 49x^2.

Now, let's consider a point where the interference is destructive, resulting in a minimum intensity. The two waves will have a phase difference of π radians (180 degrees) at this point.

The resultant amplitude at this point will be the difference of the amplitudes of the two waves: A = A1 - A2 = 3x - 4x = -x.

The resultant intensity at this point will be the square of the resultant amplitude: I_min ∝ |A|^2 = (-x)^2 = x^2.

Therefore, the ratio of intensities at maxima and minima is I_max : I_min = 49x^2 : x^2 = 49 : 1.

Hence, the correct answer is option C) 49:1.