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

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(C) The harmonic mean $H$ of two numbers $x$ and $y$ is given by $H = \frac{2xy}{x + y}$.Given $x = \frac{a}{1 - ab}$ and $y = \frac{a}{1 + ab}$.First

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$a$

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(C) The harmonic mean $H$ of two numbers $x$ and $y$ is given by $H = \frac{2xy}{x + y}$.Given $x = \frac{a}{1 - ab}$ and $y = \frac{a}{1 + ab}$.First,calculate the product $xy = \left(\frac{a}{1 - ab}\right) \left(\frac{a}{1 + ab}\right) = \frac{a^2}{1 - a^2b^2}$.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,$H = \frac{2 \left(\frac{a^2}{1 - a^2b^2}\right)}{\frac{2a}{1 - a^2b^2}} = \frac{2a^2}{2a} = a$.

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

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