If H.M. and G.M. of two positive numbers are in the ratio 12 : 13, then the numbers are in the ratio

$\frac{2 a b}{a + b} : \sqrt{a b} = 12 : 13 \Rightarrow \frac{2 \sqrt{a b}}{a + b} = \frac{12}{13}$

$\Rightarrow \sqrt{\frac{a}{b}} + \sqrt{\frac{b}{a}} = \frac{13}{6}$

$\Rightarrow \sqrt{\frac{a}{b}} = \frac{2}{3}$ or $\frac{3}{2} \Rightarrow a : b = 4 : 9$ or $9 : 4$

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