Question

The A.M. between two positive numbers a and b is twice the G. M. between them. The ratio of the numbers is

• A. $(\sqrt{2}+3):(\sqrt{2}-3)$
• B. $(2+\sqrt{3}):(2-\sqrt{3})$
• C. $(\sqrt{3}+1):(\sqrt{3}-1)$
• D. None of these

Answer 1

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

Given $2\sqrt{ab}=\frac{a+b}{2}\Rightarrow \sqrt{\frac{a}{b}}+\sqrt{\frac{b}{a}}=4\Rightarrow t^2-4t+5=0,where\sqrt{\frac{a}{b}}=t\therefore t=2\pm \sqrt{3}\Rightarrow \sqrt{\frac{a}{b}}=2\pm \sqrt{3}\therefore \frac{a}{b}=\frac{(2\pm \sqrt{3}{)}^2}{4-3}=\frac{(2\pm \sqrt{3}{)}^2}{(2{)}^2-(\sqrt{3}{)}^2}\therefore a:b=2+\sqrt{3}:2-\sqrt{3}or2-\sqrt{3}:2+\sqrt{3}$