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:

$R = \sqrt{1 + \sqrt{1 + \sqrt{1 + \sqrt{1 + . . .}}}} .$