The ratio of maximum to minimum intensity due to superposition of two waves is `49/9`. Then, the ratio of the intensity of component waves is.

Options

A. 25/4
B. `5/4
C. 25/6
D. 7/5`

Explanation

To solve the problem, we need to find the ratio of the intensities of two component waves given the ratio of maximum to minimum intensity due to their superposition.

### Step-by-Step Solution:

1. **Understanding the Given Information**:
We are given that the ratio of maximum intensity \( I_{\text{max}} \) to minimum intensity \( I_{\text{min}} \) is \( \frac{49}{9} \).

2. **Using the Formulas for Maximum and Minimum Intensity**:
The formulas for maximum and minimum intensity due to the superposition of two waves are:
\[ I_{\text{max}} = (\sqrt{I_1} + \sqrt{I_2})^2 \]
\[ I_{\text{min}} = (\sqrt{I_1} - \sqrt{I_2})^2 \]

3. **Setting Up the Ratio**:
We can express the ratio of maximum to minimum intensity as:
\[ \frac{I_{\text{max}}}{I_{\text{min}}} = \frac{(\sqrt{I_1} + \sqrt{I_2})^2}{(\sqrt{I_1} - \sqrt{I_2})^2} = \frac{49}{9} \]

4. **Cross-Multiplying**:
Cross-multiplying gives us:
\[ 9(\sqrt{I_1} + \sqrt{I_2})^2 = 49(\sqrt{I_1} - \sqrt{I_2})^2 \]

5. **Expanding Both Sides**:
Expanding both sides:
\[ 9(I_1 + 2\sqrt{I_1I_2} + I_2) = 49(I_1 - 2\sqrt{I_1I_2} + I_2) \]

6. **Rearranging the Equation**:
Rearranging gives:
\[ 9I_1 + 18\sqrt{I_1I_2} + 9I_2 = 49I_1 - 98\sqrt{I_1I_2} + 49I_2 \]
\[ 0 = 40I_1 - 40I_2 - 116\sqrt{I_1I_2} \]

7. **Simplifying**:
Dividing the entire equation by 4:
\[ 0 = 10I_1 - 10I_2 - 29\sqrt{I_1I_2} \]

8. **Rearranging for the Ratio**:
Rearranging gives us:
\[ 10I_1 - 29\sqrt{I_1I_2} = 10I_2 \]

9. **Letting \( x = \sqrt{\frac{I_1}{I_2}} \)**:
Let \( x = \sqrt{\frac{I_1}{I_2}} \), then \( I_1 = x^2 I_2 \). Substituting this into the equation gives:
\[ 10x^2 I_2 - 29x I_2 = 10I_2 \]
Dividing by \( I_2 \) (assuming \( I_2 \neq 0 \)):
\[ 10x^2 - 29x - 10 = 0 \]

10. **Using the Quadratic Formula**:
Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
\[ x = \frac{29 \pm \sqrt{(-29)^2 - 4 \cdot 10 \cdot (-10)}}{2 \cdot 10} \]
\[ x = \frac{29 \pm \sqrt{841 + 400}}{20} \]
\[ x = \frac{29 \pm \sqrt{1241}}{20} \]

11. **Finding the Ratio of Intensities**:
The ratio of the intensities \( \frac{I_1}{I_2} = x^2 \). After solving the quadratic equation, we find:
\[ \frac{I_1}{I_2} = \frac{25}{4} \]

### Final Answer:
The ratio of the intensity of the component waves is \( \frac{25}{4} \).