Question

If A.M. between $p$ and $q$ $(p\ge q)$ 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)

$(2+\sqrt{3}):(2-\sqrt{3})$

A.M between p and q is $=\frac{p+q}{2}$
and G.M between p and q is $=\sqrt{pq}$
Using Given condition,
$A.M=2.G.M\Rightarrow p+q=4\sqrt{pq}$
$\Rightarrow \frac{p}{q}-4\sqrt{\frac{p}{q}}+1=0$
$\Rightarrow \sqrt{\frac{p}{q}}=\frac{4\pm \sqrt{12}}{2}=2\pm \sqrt{3}$
But given $p\ge q$
$\therefore \sqrt{\frac{p}{q}}=2+\sqrt{3}...\left(1\right)$
Also, $\sqrt{\frac{q}{p}}=\frac{1}{2+\sqrt{3}}=\frac{1}{2+\sqrt{3}}\times \frac{2-\sqrt{3}}{2-\sqrt{3}}=2-\sqrt{3}...\left(2\right)$
Using $\left(1\right)/\left(2\right)$,we get
$\frac{p}{q}=\frac{2+\sqrt{3}}{2-\sqrt{3}}$
Hence, option 'C' is correct.