Explanation

(c)
Let the two numbers be $a$ and $b$. Then,
$\frac{AM}{GM} = \frac{\frac{a + b}{2}}{\sqrt{ab}} = \frac{m}{n}\ \ \ \lbrack given\rbrack$
$\Rightarrow \sqrt{\frac{a}{b} +}\sqrt{\frac{b}{a}} = \frac{2m}{n}$
$\Rightarrow \frac{a}{b} - 2\sqrt{\frac{a}{b}}\frac{m}{n} + 1 = 0$
$\Rightarrow x^{2} - \frac{2m}{n}x + 1 = 0,\ where\ x = \sqrt{\frac{a}{b}}$
$\Rightarrow x = \frac{m \pm \sqrt{m^{2} - n^{2}}}{n}$
$\therefore a:b = m + \sqrt{m^{2} - n^{2}}:m - \sqrt{m^{2} - n^{2}}$