Question

If A.M. of two numbers is twice of their G.M., then the ratio of greatest number to smallest number is

• A. $7-4\sqrt{3}$
• B. $7+4\sqrt{3}$
• C. 21
• D. 5

Answer 1

Answer: (B)

Let the numbers be $x$ and $y$.
Since, $\frac{x+y}{2}=2\sqrt{xy}$
$\Rightarrow x+y=4\sqrt{xy}....$(i)
$\Rightarrow (x+y{)}^2=16xy$
Also, $(x-y{)}^2=(x+y{)}^2-4xy$
$\therefore (x-y{)}^2=16xy-4xy=12xy$
$\Rightarrow x-y=2\sqrt{3xy}.....$(ii)
On solving Eqs. (i) and (ii), we get and
$x=(2+\sqrt{3})\sqrt{xy}$
and $y=(2-\sqrt{3})\sqrt{xy}$
$\therefore$ Required ratio $=\frac{x}{y}=\frac{(2+\sqrt{3})\sqrt{xy}}{(2-\sqrt{3})\sqrt{xy}}$
$=(2+\sqrt{3}{)}^2=7+4\sqrt{3}$