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?

1. With small circle radius $1$, large circle radius $x$, chord $AB=1$: by law of cosines with the large circle's $45^\circ$ arc, $1=2x^2-2x^2\cos45^\circ$.
2. $x^2=\dfrac1{2-\sqrt2}=\dfrac{2+\sqrt2}2$.
3. The answer is $\boxed{\dfrac{2+\sqrt2}2}$.