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