Q. The harmonic mean of two numbers is 4 and the arithmetic and geometric means satisfy the relation $2\ A + G^{2} = 27,$ the numbers are
A) 6, 3
B) 5, 4
C) $5,\ - 2.5$
D) $- 3,\ 1$
Explanation
(a)
Let the two numbers be $x$ and $y$, then
$A = \frac{1}{2}(x + y),\ \sqrt{xy} = G\$or $G^{2} = xy$
And $\frac{2xy}{(x + y)} = 4\ \ \Rightarrow G^{2} = 4A$
We have, $2A + G^{2} = 27\ \Rightarrow 2A + 4A = 27$
$\Rightarrow A = \frac{9}{2}$
$\Rightarrow x + y = 9$ …(i)
So, $xy = 18$ …(ii)
Solving Eqs.(i) and (ii), we get
$x = 6,\ y = 3$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.