1. Let the intensities of the two interfering beams be $I_1:I_2 = 1:9.$ So we can take $I_1 = I, \qquad I_2 = 9I.$

2. In interference, the intensities at maximum and minimum are: $I_{\max} = \left(\sqrt{I_1}+\sqrt{I_2}\right)^2,$ $I_{\min} = \left(\sqrt{I_1}-\sqrt{I_2}\right)^2.$

3. Substitute the given values: $\sqrt{I_1} = \sqrt{I}, \qquad \sqrt{I_2} = \sqrt{9I}=3\sqrt{I}.$

   Therefore, $I_{\max} = (\sqrt{I}+3\sqrt{I})^2 = (4\sqrt{I})^2 = 16I,$ $I_{\min} = (3\sqrt{I}-\sqrt{I})^2 = (2\sqrt{I})^2 = 4I.$

4. Hence the ratio of maximum to minimum intensity is $I_{\max}:I_{\min} = 16I:4I = 4:1.$

5. Checking options:

   • A: $8:1$ ❌
   • B: $4:1$ ✅
   • C: $3:1$ ❌
   • D: $9:1$ ❌

Therefore, the correct answer is Option B.