Let the intensities of the two coherent sources be in the ratio $2x:1$.

So, take $I_1 = 2xk, \qquad I_2 = k$ for some constant $k$.

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

Substitute $I_1=2xk$ and $I_2=k$: $I_{\max} = \left(\sqrt{2xk}+\sqrt{k}\right)^2$ $I_{\min} = \left(\sqrt{2xk}-\sqrt{k}\right)^2$

Now compute $I_{\max}-I_{\min}$:

Using $(a+b)^2-(a-b)^2 = 4ab$ with $a=\sqrt{2xk}$ and $b=\sqrt{k}$, $I_{\max}-I_{\min} = 4\sqrt{2xk}\sqrt{k} = 4k\sqrt{2x}.$

Compute $I_{\max}+I_{\min}$:

Using $(a+b)^2+(a-b)^2 = 2(a^2+b^2)$ $I_{\max}+I_{\min} = 2(2xk+k)=2k(2x+1).$

Therefore, $\frac{I_{\max}-I_{\min}}{I_{\max}+I_{\min}} = \frac{4k\sqrt{2x}}{2k(2x+1)}$ $= \frac{2\sqrt{2x}}{2x+1}.$

Compare with the options: $\boxed{\frac{2\sqrt{2x}}{2x+1}}$ which matches Option A.

Verification with stored answer: Stored correct answer = A, which matches our result.