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 '$x^{\prime }=\frac{a}{b}$
$4x^2+4-17x=0$
$x=4$ or $\frac{1}{4}$
$\frac{a}{b}=4$ or $\frac{a}{b}=\frac{1}{4}$.