Question

If the ratio of H. M and G.M . of two numbers are 12 :13 then prove that the numbers are in the ratio 4:9.

Answer 1

Let two number be a & b

G. M $=\sqrt{ab}$

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

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

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

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

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

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

$x+\frac{1}{x}=\frac{13}{6}$

$=6x^2-13x+6=0$

$\Rightarrow x=2/3$ or $3/2$

So, $x^2$=ratio of number

$=4/9$ or $9/4$