Q. If the AM of two numbers be $A$ and GM be $G$, then the numbers will be
A) $A \pm (A^{2} - G^{2})$
B) $\sqrt{A} \pm \sqrt{A^{2} - G^{2}}$
C) $A \pm \sqrt{(A + G)(A - G)}$
D) $\frac{A \pm \sqrt{(A + G)(A - G)}}{2}$
Explanation
(c)
Let the two numbers be $a$ and $b$
$\therefore$ AM = $\frac{a + b}{2} = A$ …(i)
and $GM = \sqrt{ab}\ \ \ \Rightarrow \ \ \ G^{2} = ab$ …(ii)
Now, $(a - b)^{2} = (a + b)^{2} - 4ab = (2A)^{2} - 4G^{2}$
$= 4(A^{2} - G^{2})$
$\Rightarrow a - b = \pm 2\sqrt{(A + G)(A - G)}$ ...(iii)
On solving Eqs. (i) and (iii), we get
$a = A \pm \sqrt{(A + G)(A - G)}$
and $b = A \pm \sqrt{(A + G)(A - G)}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.