Let the radius of the bigger circle and the smaller circle be $\(R \& r\)$ respectively.
We are given that the area of the bigger circle is $\(\pi=\pi R^2 \Rightarrow R=\sqrt{1}=1\)$ $\(\qquad\)$
Also it is known that the area of the bigger circle to the smaller circle is in ratio $\(2: 1\)$, so we get

$$
\(\pi R^2: \pi r^2=2: 1 \Rightarrow R: r=\sqrt{2}: 1\)
$$


Using (1) \& (2), we get $\(\frac{R}{r}=\frac{\sqrt{2}}{1}=\frac{1}{r} \Rightarrow r=\frac{1}{\sqrt{2}}\)$
So, the distance between the centers of the two circles is sum of their radii

$$
\(=\mathrm{R}+\mathrm{r}=1+\frac{1}{\sqrt{2}}=\frac{\sqrt{2}+1}{\sqrt{2}}\)
$$


Hence the answer is (B).