Challenge Level

1) An ellipse with semi axes $a$ and $b$ fits between two circles of radii $a$ and $b$ (where $b> a$) as shown in the diagram. If the area of the ellipse is equal to the area of the annulus what is the ratio $b:a$?

(2) Find the value of $R$ if this sequence of 'nested square roots' continues indefinitely: $$R=\sqrt{1 + \sqrt{1 + \sqrt {1 + \sqrt {1 + ...}}}}.$$