(1) The areas of the ellipse and the annulus are

π ab

and

π (b^{2} - a^{2})

and so

\begin{matrix} b^{2} - a^{2} & = ab \\ (b / a)^{2} - (b / a) - 1 & = 0 \end{matrix}

and so, as

b / a

is positive,

b / a = (1 + \sqrt{5}) / 2

which is the Golden Ratio.

(2)As

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

is an infinite 'nested sequence' we have

$R^{2} - 1 = R$

so we find the value of $R$ by solving the equation $R^{2} - R - 1 = 0$ and, as $R$ must be positive, $R = (1 + \sqrt{5}) / 2$ , the Golden Ratio.