Question

The arithmetic mean between two distinct positive numbers is twice the geometric mean between them. Find the ratio of greater to smaller.

Answer 1

Let two nor be a and b

$\therefore A.P=\frac{a+b}{2}$

$C.P=\sqrt{ab}$

$\therefore \frac{a+b}{2}=2\sqrt{ab}$

$\Rightarrow a+b=4\sqrt{ab}$

$\Rightarrow (a+b{)}^2=16ab$

$\Rightarrow a^2+2ab+b^2=16ab$

$\Rightarrow a^2+b^2=14ab$

$\Rightarrow \frac{a}{b}+\frac{b}{a}=14$

Let $\frac{b}{a}=x$

$\therefore x+\frac{1}{x}=14$

$\Rightarrow x^2+1=14x$

$\Rightarrow x^2-14x+1=0$

$\therefore x=\frac{14\pm \sqrt{(4{)}^2-4}}{2}$

$=\frac{14\pm \sqrt{192}}{2}$

$=\frac{14\pm 2\sqrt{48}}{2}$

$=7\pm \sqrt{48}$

$\therefore$ Ratio of greater to smaller $=\frac{b}{a}$

$=7+\sqrt{48}$

$=7+4\sqrt{3}$