If $A$ be the A.M. and $H$ the H.M. between two numbers $a$ and $b$ , then show
$\frac{a - A}{a - H} \times \frac{b - A}{b - H} = \frac{A}{H} .$

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Solution

$A$ is $A . M$ of $a, b$ then $A = \frac{a + b}{2}$
$H$ is $H . M$ of $a, b$ then $H = \frac{2 a b}{a + b}$
Now subettuting this in
$\frac{a - A}{a - H} \times \frac{b - A}{b - H} = \frac{a - \frac{a + b}{2}}{a - \frac{2 a b}{a + b}} \times \frac{b - \frac{a + b}{2}}{b - \frac{2 a b}{a + b}}$
$= \frac{a - b}{2 (a^{2} - a b)} \times \frac{(b - a)}{2 (b^{2} - a b)} = \frac{1}{2 a} \times \frac{1}{2 b}$
$= \frac{1}{4 a b} = \frac{(\frac{a + b}{2})}{\frac{2 a b}{a + b}} = \frac{A}{H}$

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