Question

If A.M. between positive numbers $p$ and $q(p \geq q)$ is two times the $G M$, then $p$ : $q$ is -

• A. $1: 1$
• B. $2: 1$
• C. $(2+\sqrt{3}):(2-\sqrt{3})$
• D. $3: 1$

Answer 1

Answer: (C)

Given numbers are $p$ and $q$
$ \text { A.M. }=\frac{p+q}{2}, G M=\sqrt{p q}$
$ =2 \text { (Given) }$
$ \therefore \frac{p+q}{2 \sqrt{p q}}=2$
$\therefore \frac{p+q+2 \sqrt{p q}}{p+q-2 \sqrt{p q}}=\frac{3}{1}$
(componendo dividendo method)
$\frac{(\sqrt{p}+\sqrt{q})^2}{(\sqrt{p}-\sqrt{q})^2}=\frac{3}{1} $
$\Rightarrow \frac{\sqrt{p}+\sqrt{q}}{\sqrt{p}-\sqrt{q}}=\sqrt{3}$
$\therefore \frac{2 \sqrt{p}}{2 \sqrt{q}}=\frac{\sqrt{3}+1}{\sqrt{3}-1}$
$\therefore \frac{p}{q}=\left(\frac{\sqrt{3}+1}{\sqrt{3}-1}\right)^2=\frac{4+2 \sqrt{3}}{4-2 \sqrt{3}}=\frac{2+\sqrt{3}}{2-\sqrt{3}}$