Question

If two positive numbers are in the ratio $3+2\sqrt{2};3-2\sqrt{2},$ then the ratio between their $A.M.$ and $G.M.$ is

• A. 6 : 1
• B. 3 : 2
• C. 2 : 1
• D. 3 : 1
• E. 1 : 6

Answer 1

Answer: (D)

Let two positive numbers be $a$ and $b$.
$\therefore \frac{a}{b}=\frac{3+2\sqrt{2}}{3-2\sqrt{2}}\times \frac{3+2\sqrt{2}}{3+2\sqrt{2}}=\frac{9+8+12\sqrt{2}}{9-8}$
$=17+12\sqrt{2}$
$\therefore \frac{AM}{GM}=\frac{\frac{a+b}{2}}{\sqrt{ab}}=\frac{(17+12\sqrt{2})b+b}{2\sqrt{(17+12\sqrt{2})b^2}}$
$=\frac{18+12\sqrt{2}}{2\sqrt{17+12\sqrt{2}}}$
$=\frac{9+6\sqrt{2}}{\sqrt{(3+2\sqrt{2}{)}^2}}=\frac{3(3+2\sqrt{2})}{3+2\sqrt{2}}$
$=\frac{3}{1}$
$\therefore AM:GM=3:1$