Question

The A.M. between two positive numbers a and b is twice the G. M. between them. The ratio of the numbers is

• A. $(\sqrt{2}+3):(\sqrt{2}-3)$
• B. $(2+\sqrt{3}):(2-\sqrt{3})$
• C. $(\sqrt{3}+1):(\sqrt{3}-1)$
• D. None of these

Answer 1

The correct option is B $(2+\sqrt{3}):(2-\sqrt{3})$

Given $2\sqrt{ab}=\frac{a+b}{2}$
$\Rightarrow \frac{a+b}{2\sqrt{ab}}=\frac{2}{1}$
$\Rightarrow \frac{a+b+2\sqrt{ab}}{a+b-2\sqrt{ab}}=\frac{2+1}{2-1}$
$\Rightarrow \frac{(\sqrt{a}+\sqrt{b}{)}^2}{(\sqrt{a}-\sqrt{b}{)}^2}=\frac{{\sqrt{3}}^2}{1^2}$
$\Rightarrow \frac{\sqrt{a}+\sqrt{b}}{\sqrt{a}-\sqrt{b}}=\frac{\sqrt{3}+1}{\sqrt{3}-1}$
$\Rightarrow \frac{\sqrt{a}}{\sqrt{b}}=\frac{\sqrt{3}+1}{\sqrt{3}-1}$
$\Rightarrow (\frac{\sqrt{a}}{\sqrt{b}}{)}^2=(\frac{\sqrt{3}+1}{\sqrt{3}-1}{)}^2$
$\Rightarrow \frac{a}{b}=\frac{2+\sqrt{3}}{2-\sqrt{3}}$