Question

If the AM of two positive numbers be two times their geometric mean then the ratio of the numbers is

• A. $2+\sqrt{3}:2-\sqrt{3}$
• B. $7+4\sqrt{3}:7-4\sqrt{3}$
• C. $2:7+4\sqrt{3}$
• D. $2:\sqrt{3}$

Answer 1

The correct option is A $2+\sqrt{3}:2-\sqrt{3}$

Let $A$ be the $AM$ and $G$ be the $GM$ of $a$ and $b$ .
Then the numbers $a$ and $b$ are roots of the equation
$x^2-2Ax+G^2=0\Rightarrow x^2-4Gx+G^2=0(\because A=2G)$
$\Rightarrow x=\frac{4G\pm \sqrt{16G^2-4G^2}}{2}$
$\Rightarrow x=\frac{4G\pm 2\sqrt{3}G}{2}=2\pm \sqrt{3}$
Hence $a:b=2+\sqrt{3}:2-\sqrt{3}$