The given question asks us to find the ratio of the intensity at the center of a bright fringe to the intensity at a point one-quarter of the distance between two fringes from the center in an interference pattern produced by two identical coherent waves.

Interference in Coherent Waves:
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When two identical coherent waves superpose, they produce an interference pattern. In an interference pattern, bright and dark fringes are formed due to constructive and destructive interference, respectively.

Intensity in Interference Pattern:
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The intensity of a wave is directly proportional to the square of its amplitude. In an interference pattern, the intensity is maximum at the bright fringes and minimum at the dark fringes.

Finding the Intensity Ratio:
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To find the ratio of the intensity at the center of a bright fringe to the intensity at a point one-quarter of the distance between two fringes from the center, we need to consider the nature of the interference pattern.

1. Intensity at the Center of a Bright Fringe:
- At the center of a bright fringe, the waves from both sources are in phase and undergo constructive interference.
- This results in the maximum intensity at the center of the bright fringe.

2. Intensity at a Point One-Quarter Distance from the Center:
- At a point one-quarter of the distance between two fringes from the center, the waves from the two sources are out of phase by 180 degrees.
- This leads to destructive interference, causing the intensity to be minimum at this point.

3. Intensity Ratio Calculation:
- The intensity at the center of a bright fringe is maximum, so let's assign it a value of 1.
- The intensity at a point one-quarter distance from the center is minimum and can be considered 0.
- Therefore, the ratio of the intensity at the center to the intensity at the one-quarter distance point is 1/0, which is undefined.

Conclusion:
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The given answer of '2' is incorrect, and the ratio of the intensity at the center of a bright fringe to the intensity at a point one-quarter of the distance between two fringes from the center is undefined. This is because the intensity at the one-quarter distance point is zero due to destructive interference.