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