1. Let the intensities of the two coherent sources be $I_1:I_2=1:4$ So, take $I_1=I,\, I_2=4I$

2. For interference of two coherent sources: $I_{\max}=(\sqrt{I_1}+\sqrt{I_2})^2$ $I_{\min}=(\sqrt{I_1}-\sqrt{I_2})^2$

3. Compute these values: $\sqrt{I_1}=\sqrt{I},\qquad \sqrt{I_2}=2\sqrt{I}$

   Hence, $I_{\max}=(\sqrt{I}+2\sqrt{I})^2=(3\sqrt{I})^2=9I$ $I_{\min}=(2\sqrt{I}-\sqrt{I})^2=(\sqrt{I})^2=I$

4. Now evaluate \frac{I_{\max}+I_{\min}}{I_{\max}-I_{\min}}=rac{9I+I}{9I-I}=rac{10I}{8I}=\frac{5}{4}

5. According to the question, $\frac{I_{\max }+I_{\min }}{I_{\max }-I_{\min }}=\frac{2\alpha+1}{\beta+3}$ So, $\frac{2\alpha+1}{\beta+3}=\frac{5}{4}$

6. Cross-multiply: $4(2\alpha+1)=5(\beta+3)$ $8\alpha+4=5\beta+15$ $8\alpha-5\beta=11$

7. Now test the options for $\frac{\alpha}{\beta}$.

   • If $\frac{\alpha}{\beta}=\frac{3}{2}$, let $\alpha=3k,\beta=2k$: $8(3k)-5(2k)=24k-10k=14k=11$ Not convenient for integer-like matching.

   • If $\frac{\alpha}{\beta}=2$, let $\alpha=2,\beta=1$: $\frac{2\alpha+1}{\beta+3}=\frac{5}{4}$ This matches exactly.

   • If $\frac{\alpha}{\beta}=\frac{1}{2}$, let $\alpha=1,\beta=2$: $\frac{2\alpha+1}{\beta+3}=\frac{3}{5}$ Not matching.

   • If $\frac{\alpha}{\beta}=1$, let $\alpha=\beta=1$: $\frac{2\alpha+1}{\beta+3}=\frac{3}{4}$ Not matching.

8. Therefore, $\frac{\alpha}{\beta}=2$

So the correct option is B.