Question

Let $G$ be the geometric mean of two positive numbers $a$ and $b$, and $M$ be the arithmetic mean of $\frac{1}{a}$ and $\frac{1}{b}$. If $\frac{1}{M}:G$ is $4:5,$ , then $a : b$ can be :

• A. 1 : 4
• B. 1 : 2
• C. 2 : 3
• D. 3 : 4

Answer 1

Answer: (A)

$G = \sqrt{ab}, M = \frac{a + b}{2ab}$
$MG \times G = \frac{a + b}{2}$
$\frac{1}{MG} = \frac{4}{5}$
$\sqrt{ab} = \frac{2}{5}(a + b)$
$ab = \frac{4}{25} (a^2 + b^2 + 2ab)$
$4 \frac{a^2}{b^2} + 4 - 17 \frac{a}{b} \Rightarrow 0$
Let '$x' = \frac{a}{b}$
$4x^2 + 4 - 17 x = 0$
$x = 4$ or $\frac{1}{4}$
$\frac{a}{b} = 4$ or $\frac{a}{b} = \frac{1}{4}$.