Question

The sum of the $A.M$. and $G.M$. of two positive numbers is equal to the difference between the numbers. The numbers are in the ratio

• A. $1 : 3$
• B. $1 : 6$
• C. $ 9 : 1$
• D. $1 : 12$

Answer 1

Answer: (C)

$\frac{a+b}{2} +\sqrt{ab} = a-b $
$\Rightarrow 2\sqrt{ab} = a-3b $
$ \Rightarrow 4ab = \left(a-3b\right)^{2}$
$\Rightarrow a^{2} -10ab +9b^{2} = 0$
$ \Rightarrow \left(a-9b\right)\left(a-b\right) = 0$
$ \Rightarrow a= 9b \left[\because a\ne b\right]$
$\Rightarrow \frac{a}{b} = \frac{9}{1} $