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

Answer: (B)

Given $2\sqrt{ab}=\frac{a+b}{2}\Rightarrow \sqrt{\frac{a}{b}}+\sqrt{\frac{b}{a}}=4$
$\Rightarrow t^2-4t+5=0$, where $\sqrt{\frac{a}{b}}=t$
$\therefore t=2\pm \sqrt{3}$
$\Rightarrow \sqrt{\frac{a}{b}}=2\pm \sqrt{3}$
$\therefore \frac{a}{b}=\frac{(2\pm \sqrt{3}{)}^2}{4-3}=\frac{(2\pm \sqrt{3}{)}^2}{(2{)}^2-(\sqrt{3}{)}^2}$
$\therefore a:b=2+\sqrt{3}:2-\sqrt{3}$
or $2-\sqrt{3}:2+\sqrt{3}$