smc

Let the radius of the larger and smaller circles be $x$ and $y$, respectively. Also, let their centers be $O_1$ and $O_2$, respectively. Then the ratio we need to find is $$\frac{\pi x^2}{\pi y^2}=\frac{x^2}{y^2}$$ Draw the radii from the centers of the circles to $A$ and $B$. We can easily conclude that the $30^{\circ }$ belongs to the larger circle, and the $60$ degree arc belongs to the smaller circle. Therefore, $m\angle A{O}_{1}B=30^{\circ }$ and $m\angle A{O}_{2}B=60^{\circ }$. Note that $\Delta A{O}_{2}B$ is equilateral, so when chord AB is drawn, it has length $y$. Now, applying the Law of Cosines on $\Delta A{O}_{1}B$: $$y^2=x^2+x^2-2x^2\cos 30=2x^2-x^2\sqrt{3}=(2-\sqrt{3})x^2$$ $$\frac{x^2}{y^2}=\frac{1}{2-\sqrt{3}}=\frac{2+\sqrt{3}}{4-3}=2+\sqrt{3}=\boxed{2+\sqrt{3}}$$