Question

The arithmetic mean of two positive numbers $a$ and $b$ exceeds their geometric mean by $2$ and the harmonic mean is one-fifth of the greater of $a$ and $b$ , such that $\alpha =a+b$ and $\beta =\left|a-b\right|$ , then the value of $\alpha +{\beta }^2$ is equal to

• A. $96$
• B. $234$
• C. $74$
• D. $84$

Answer 1

Answer: (C)

$A=G+2$
$H=\frac{b}{5}$
(let $b>a)$
$G^2=AH\Rightarrow ab=\left(\frac{a+b}{2}\right)\frac{y}{5}\Rightarrow 10a=a+b\Rightarrow b=9a$
$A=\frac{a+b}{2}=5a,G=\sqrt{9a\cdot a}=3a$
$\Rightarrow 5a=3a+2\Rightarrow a=1\Rightarrow b=9$
$\Rightarrow \alpha =10$ and $\beta =8$
$\Rightarrow \alpha +{\beta }^2=74$