Question

If the arithmetic means of two positive numbers $ab(a>b)$ is twice their G.M., then $a:b$ is

• A. $6+\sqrt{7}:6-\sqrt{7}$
• B. $2+\sqrt{3}:2-\sqrt{3}$
• C. $5+\sqrt{6}:5-\sqrt{6}$
• D. None of these

Answer 1

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

(As $a>b$)
Let $(A.M.=\frac{a+b}{2},G.M.=\sqrt{ab})$
$\therefore$ The equation whose roots a and b is
$x^2-(a+b)x+ab=0$
$x^2-2Ax+G^2=0$
$x^2-2\left(2G\right)x+G^2=0$ $GivenA.M=2G.M$
$x^2-4Gx+G^2=0$ $\therefore \left(\frac{a+b}{2}\right)=2G.M$
$\therefore x=\frac{4G\pm 2\sqrt{3}.G}{2}$ $a+b=4G.M=4G$
$=(2\pm \sqrt{3})G$
$\therefore \frac{a}{b}=\frac{2+\sqrt{3}}{2-\sqrt{3}}$