If `I_1/I_2` =25 then find the value of `(I_max - I_min)/ (I_max+I_min)`

Answer

To solve the problem, we need to find the value of \((I_{\text{max}} - I_{\text{min}}) / (I_{\text{max}} + I_{\text{min}})\) given that the ratio of intensities \(I_1/I_2 = 25\).

### Step-by-Step Solution:

1. **Identify the Intensities**:
Given the ratio \(I_1/I_2 = 25\), we can assign:
\[ I_1 = 25k \quad \text{and} \quad I_2 = k \]
for some constant \(k\).

2. **Calculate \(I_{\text{max}}\)**:
The formula for maximum intensity \(I_{\text{max}}\) is given by:
\[ I_{\text{max}} = (\sqrt{I_1} + \sqrt{I_2})^2 \]
Substituting the values of \(I_1\) and \(I_2\):
\[ I_{\text{max}} = (\sqrt{25k} + \sqrt{k})^2 = (5\sqrt{k} + \sqrt{k})^2 = (6\sqrt{k})^2 = 36k \]

3. **Calculate \(I_{\text{min}}\)**:
The formula for minimum intensity \(I_{\text{min}}\) is given by:
\[ I_{\text{min}} = (\sqrt{I_1} - \sqrt{I_2})^2 \]
Substituting the values of \(I_1\) and \(I_2\):
\[ I_{\text{min}} = (\sqrt{25k} - \sqrt{k})^2 = (5\sqrt{k} - \sqrt{k})^2 = (4\sqrt{k})^2 = 16k \]

4. **Calculate \(I_{\text{max}} - I_{\text{min}}\)**:
Now, we find \(I_{\text{max}} - I_{\text{min}}\):
\[ I_{\text{max}} - I_{\text{min}} = 36k - 16k = 20k \]

5. **Calculate \(I_{\text{max}} + I_{\text{min}}\)**:
Next, we find \(I_{\text{max}} + I_{\text{min}}\):
\[ I_{\text{max}} + I_{\text{min}} = 36k + 16k = 52k \]

6. **Calculate the Final Ratio**:
Now we can find the ratio:
\[ \frac{I_{\text{max}} - I_{\text{min}}}{I_{\text{max}} + I_{\text{min}}} = \frac{20k}{52k} = \frac{20}{52} = \frac{5}{13} \]

### Final Answer:
\[ \frac{I_{\text{max}} - I_{\text{min}}}{I_{\text{max}} + I_{\text{min}}} = \frac{5}{13} \]