Question

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

• A. $1:2$
• B. $2:3$
• C. $3:4$
• D. $Noneofthese$

Answer 1

The correct option is D $Noneofthese$

Given,

$H.M.=\frac{2ab}{a+b}G.M.=\sqrt{ab}$

$\frac{H.M.}{G.M.}=\frac{2ab}{(a+b)\sqrt{ab}}=\frac{12}{13}$

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

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

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

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

Let $\sqrt{\frac{a}{b}}=x$

Then,

$x+\frac{1}{x}=\frac{13}{6},{6x}^2+6=13x,{6x}^2-13x+6=0,{6x}^2-9x-4x+6=0,(3x-2)(2x-3)=0.$

$x=\frac{2}{3},\frac{3}{2}$

$x^2=\frac{4}{9},\frac{9}{4}$

$\frac{a}{b}=\frac{4}{9},\frac{9}{4}$

$\therefore$ None of these.