The harmonic mean of frac{a}{1 - ab} and frac | vedclass

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(C) The harmonic mean $(H.M.)$ of two numbers $x$ and $y$ is given by the formula $H.M. = \frac{2xy}{x + y}$.Here,$x = \frac{a}{1 - ab}$ and $y = \fra

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$a$

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(C) The harmonic mean $(H.M.)$ of two numbers $x$ and $y$ is given by the formula $H.M. = \frac{2xy}{x + y}$.Here,$x = \frac{a}{1 - ab}$ and $y = \frac{a}{1 + ab}$.First,calculate the product $xy = \left(\frac{a}{1 - ab}ight) \left(\frac{a}{1 + ab}ight) = \frac{a^2}{1 - a^2b^2}$.Next,calculate the sum $x + y = \frac{a}{1 - ab} + \frac{a}{1 + ab} = \frac{a(1 + ab) + a(1 - ab)}{(1 - ab)(1 + ab)} = \frac{a + a^2b + a - a^2b}{1 - a^2b^2} = \frac{2a}{1 - a^2b^2}$.Now,substitute these into the $H.M.$ formula:$H.M. = \frac{2 \left(\frac{a^2}{1 - a^2b^2}ight)}{\frac{2a}{1 - a^2b^2}} = \frac{2a^2}{1 - a^2b^2} 	imes \frac{1 - a^2b^2}{2a} = \frac{2a^2}{2a} = a$.

The harmonic mean of $\frac{a}{1 - ab}$ and $\frac{a}{1 + ab}$ is

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