Question

The harmonic mean of two positive numbers $a$ and $b$ is $4,$ their arithmeitc mean is $A$ and the geometric mean is $G.$ If $2A+G^{2}=27,$ $a+b=\alpha $ and $\left|a - b\right|=\beta $ , then the value of $\frac{\alpha }{\beta }$ is equal to

• A. $1$
• B. $3$
• C. $\frac{5}{2}$
• D. $5$

Answer 1

Answer: (B)

Given, harmonic mean $H=4$
We know that, $G^{2}=AH$
Since, $2A+G^{2}=27$
$\Rightarrow 2A+AH=27$
$\Rightarrow 2A+4A=27$
$\Rightarrow A=\frac{27}{6}=\frac{9}{2}=\frac{a + b}{2}\Rightarrow a+b=9$
$G^{2}=AH=\frac{9}{2}\times 4=18$
$\Rightarrow ab=18$
$\left|a - b\right|=\sqrt{\left(a + b\right)^{2} - 4 a b}=\sqrt{81 - 4 \times 18}=3$
$\Rightarrow \alpha =9,\beta =3$
$\Rightarrow \frac{\alpha }{\beta }=3$