Question

The harmonic mean of two numbers is $4$ .Their arithmetic mean is A and geometric mean is G. If G satisfies $2A+G^2=27$, the numbers are

• A. $1,13$
• B. $9,12$
• C. $3,6$
• D. $4,8$

Answer 1

Answer: (C)

$3,6$

Let the two numbers be $a$ and $b$

Given: $G^2=27-2A$
We know that $H=\frac{G^2}{A}$
$\Rightarrow 4=\frac{(27-2A)}{A}$
On solving, we get

$A=\frac{9}{2}$ and $G=3\sqrt{2}$

$\Rightarrow \frac{a+b}{2}=\frac{9}{2}$ and $\sqrt{ab}=3\sqrt{2}$

$\Rightarrow a+b=9$ .... $\left(1\right)$ and $ab=18$ ..... $\left(2\right)$

Multiply equation $\left(1\right)$ by $b$ and substitute $\left(2\right)$ in $\left(1\right),$ we get

$b^2-9b+18=0$

$\Rightarrow (b-3)(b-6)=0$

$\Rightarrow b=3,6$ and $a=6,3$

Hence, the numbers are $3,6$.