Explanation

(b)
Let the numbers be $x$ and $y$
Since, $\frac{x + y}{2} = 2\sqrt{xy}$
$\Rightarrow x + y = 4\sqrt{xy}$
$\Rightarrow (x + y)^{2} = 16xy$ …(i)
Also $(x - y)^{2} = (x + y)^{2} - 4xy$
$\therefore\ (x - y)^{2} = 16xy - 4xy = 12xy$
$\Rightarrow \ x - y = 2\sqrt{3xy}$ …(ii)
On solving Eqs. (i) and (ii), we get
$x = (2 + \sqrt{3})\sqrt{xy}$
and $y = (2 - \sqrt{3})\sqrt{xy}$
$\therefore$ Required ratio = $\frac{x}{y} = \frac{(2 + \sqrt{3})\sqrt{xy}}{(2 - \sqrt{3})\sqrt{xy}}$
$= \left( 2 + \sqrt{3} \right)^{2} = 7 + 4\sqrt{3}$