Q. If the ratio of AM between two positive real numbers $a$ and $b$ to their HM is $m\ :n,$ then $a:b$ is equal to
A) $\frac{\sqrt{(m - n)} + \sqrt{n}}{\sqrt{m - n} - \sqrt{n}}$
B) $\frac{\sqrt{n} + \sqrt{m - n}}{\sqrt{n} - \sqrt{m - n}}$
C) $\frac{\sqrt{m} + \sqrt{m - n}}{\sqrt{m} - \sqrt{m - n}}$
D) $\frac{\sqrt{(m - n)} + \sqrt{m}}{\sqrt{m - n} - \sqrt{m}}$
Explanation
(c)
Let $A = m\lambda,\ H = n\lambda$
$\therefore\ G^{2} = AH = mn\lambda^{2}$
Also, $a$ and $b$ be the roots of
$x^{2} - (a + b)x + ab = 0$
$\Rightarrow \ x^{2} - 2m\lambda x + mn\lambda^{2} = 0$
$\Rightarrow \ x = \lambda\sqrt{m}\ \left\{ \sqrt{m} \pm \sqrt{m - n} \right\}$
$\therefore\ a:b = \left( \sqrt{m} + \sqrt{m - n} \right):\left( \sqrt{m} - \sqrt{m - n} \right)$
or $\left( \sqrt{m} - \sqrt{m - n} \right)\ :\ \left( \sqrt{m} + \sqrt{m - n} \right)$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.