Assertion: Ratio of maximum intensity and minimum intensity in interference is `25:1`. The amplitudes ratio of two waves should be `3:2`.
Reason: `I_(max)/I_(min) = ((A_1+A_2)/(A_1-A_2))^2` .

Options

A. If both Assertion and Reason are true and the Reason is correct explanation of the Assertion.
B. If both Assertion and Reason are true but Reason is not the correct explanation of Assertion.
C. If Assertion is true, but the Reason is false.
D. If Assertion is false but the Reason is true.

Answer

To solve the given problem, we need to analyze the assertion and reason provided, using the formulas for maximum and minimum intensity in the context of wave interference.

### Step-by-Step Solution:

1. **Understanding the Assertion**:
The assertion states that the ratio of maximum intensity \( I_{max} \) to minimum intensity \( I_{min} \) is \( 25:1 \). It also claims that the ratio of the amplitudes of two waves is \( 3:2 \).

2. **Using the Formulas**:
The formulas for maximum and minimum intensity in interference are:
\[ I_{max} = (A_1 + A_2)^2 \]
\[ I_{min} = (A_1 - A_2)^2 \]

3. **Setting Up the Ratio**:
The ratio of maximum intensity to minimum intensity can be expressed as:
\[ \frac{I_{max}}{I_{min}} = \frac{(A_1 + A_2)^2}{(A_1 - A_2)^2} \]

4. **Substituting the Amplitude Ratio**:
Given the amplitude ratio \( \frac{A_1}{A_2} = \frac{3}{2} \), we can express \( A_1 \) and \( A_2 \) in terms of a common variable. Let:
\[ A_2 = 2k \quad \text{and} \quad A_1 = 3k \]
where \( k \) is a constant.

5. **Calculating \( I_{max} \) and \( I_{min} \)**:
Substitute \( A_1 \) and \( A_2 \) into the formulas:
\[ I_{max} = (3k + 2k)^2 = (5k)^2 = 25k^2 \]
\[ I_{min} = (3k - 2k)^2 = (1k)^2 = k^2 \]

6. **Finding the Ratio**:
Now, we can find the ratio:
\[ \frac{I_{max}}{I_{min}} = \frac{25k^2}{k^2} = 25 \]
Therefore, the ratio \( I_{max} : I_{min} = 25 : 1 \).

7. **Conclusion about the Assertion**:
Since we have shown that the ratio of maximum intensity to minimum intensity is indeed \( 25:1 \) when the amplitude ratio is \( 3:2 \), the assertion is true.

8. **Verifying the Reason**:
The reason states that \( \frac{I_{max}}{I_{min}} = \left(\frac{A_1 + A_2}{A_1 - A_2}\right)^2 \). We derived this formula earlier, confirming that it is correct.

9. **Final Conclusion**:
Both the assertion and the reason are true, and the reason correctly explains the assertion.