Q. If $a$ is positive and if $A$ and $G$ are the arithmetic mean and the geometric mean of the roots of $x^{2} - 2ax + a^{2} = 0$ respectively, then
A) $A = G$
B) $A = 2G$
C) $2A = G$
D) $A^{2} = G$
Explanation
(a)
Let $\alpha$ and $\beta$ are the roots of the equation $x^{2} - 2ax + a^{2} = 0$
$\therefore\ \ \ \ \alpha + \beta = 2a$ and $\alpha\beta = a^{2}$ …(i)
Since, $A = \frac{\alpha + \beta}{2}$ and $G = \sqrt{\alpha\beta}$
$\Rightarrow A = a$ and $G^{2} = a^{2}$ [from Eq. (i)]
$\Rightarrow G^{2} = A^{2}\ \Rightarrow G = A$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.