Two circles intersect at points $A$ and $B$. The minor arcs $AB$ measure $30^\circ$ on one circle and $60^\circ$ on the other circle. What is the ratio of the area of the larger circle to the area of the smaller circle? $\textbf{(A) }2\qquad$ $ \textbf{(B) }1+\sqrt3\qquad$ $ \textbf{(C) }3\qquad$ $ \textbf{(D) }2+\sqrt3\qquad$ $ \textbf{(E) }4\qquad$
Solution
Assume the small circle has radius 1, and the large circle has radius $x$.
In the small circle the $60^\circ$ arc $AB$ cuts off an equilateral triangle, so $AB=1$. In the large circle (center $O$) $AO=BO=x$ and $\angle AOB=30^\circ$.
Two circles intersect at $A,B$. The minor arc $AB$ measures $45^\circ$ on one circle and $60^\circ$ on the other. What is the ratio of the larger circle's area to the smaller's?
