If the ratio of H.M. and G.M. of two quantiti | vedclass

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(D) Given that $\frac{H.M.}{G.M.} = \frac{12}{13}$.We know that $H.M. = \frac{2ab}{a+b}$ and $G.M. = \sqrt{ab}$.So,$\frac{2ab}{(a+b)\sqrt{ab}} = \frac

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None of these

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(D) Given that $\frac{H.M.}{G.M.} = \frac{12}{13}$.We know that $H.M. = \frac{2ab}{a+b}$ and $G.M. = \sqrt{ab}$.So,$\frac{2ab}{(a+b)\sqrt{ab}} = \frac{12}{13} \Rightarrow \frac{2\sqrt{ab}}{a+b} = \frac{12}{13}$.This implies $\frac{a+b}{2\sqrt{ab}} = \frac{13}{12}$.Using componendo and dividendo,$\frac{(a+b)+2\sqrt{ab}}{(a+b)-2\sqrt{ab}} = \frac{13+12}{13-12} = \frac{25}{1}$.$\frac{(\sqrt{a}+\sqrt{b})^2}{(\sqrt{a}-\sqrt{b})^2} = 25$.Taking the square root,$\frac{\sqrt{a}+\sqrt{b}}{\sqrt{a}-\sqrt{b}} = 5$.Again,using componendo and dividendo,$\frac{2\sqrt{a}}{2\sqrt{b}} = \frac{5+1}{5-1} = \frac{6}{4} = \frac{3}{2}$.Squaring both sides,$\frac{a}{b} = \frac{9}{4}$.Thus,the ratio is $9:4$,which is not among the given options.

If the ratio of $H.M.$ and $G.M.$ of two quantities is $12:13$,then the ratio of the numbers is

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