A Geometery

Let $O$ be the center of the pink circle, $C$ be the center of the yellow circle and $P$ be the point of tangency of the yellow circle to radial segment $OB$ . Also, let the radius $OB$ of the pink circle be $R$ .

Letting the radius $CP$ of the yellow circle be $r$ , we have that triangle $\Delta OCP$ is a right triangle with $\angle COP = 30^{\circ}$ and hypotenuse $OC = R - r$ . Therefore

$\sin(30^{\circ}) = \frac{r}{R - r} \Longrightarrow \frac{1}{2} = \frac{r}{R - r} \Longrightarrow R - r = 2r \Longrightarrow \frac{R}{r} = 3$ .

Now the yellow area is $\pi r^{2}$ and the pink area is $\frac{1}{2}*R^{2}*\frac{\pi}{3} - \pi r^{2}$ , and thus the ratio of the pink area to the yellow area is

$\dfrac{\frac{1}{2}*R^{2}*\frac{\pi}{3} - \pi r^{2}}{\pi r^{2}} = \frac{1}{6}*(\frac{R}{r})^{2} - 1 = \frac{3^{2}}{6} - 1 = \frac{1}{2}$ .

Thus $a + b = 1 + 2 = \boxed{3}$ .