The harmonic mean of a/(1 - ab) and a/(1 + ab) is

Question

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

(a) a 

(b) \(\frac{1}{1-a^2b^2}\)

(c) \(\frac{a}{1-a^2b^2}\)

(d) \(\frac{a}{\sqrt{1-a^2b^2}}\)

Answer 1

(a) a

∵ Harmonic mean of two quantities a and b is \(\frac{2ab}{a+b}\)

∴ Reqd. H.M. = \(\frac{2\times\big(\frac{a}{1-ab}\big)\times\big(\frac{a}{1+ab}\big)}{\frac{a}{1-ab}+\frac{a}{1+ab}}\)

= \(\frac{\frac{2a^2}{(1-a^2b^2)}}{\frac{a+a^2b+a-a^2b}{(1-a^2b^2)}}\) = \(\frac{2a^2}{a}\)= a.