Question

If $\frac{1}{a} , \frac{1}{b} , \frac{1}{c} $ are in A. P., then $\left(\frac{1}{a} + \frac{1}{b} - \frac{1}{c}\right) \left(\frac{1}{b} + \frac{1}{c} - \frac{1}{a}\right) $ is equal to

• A. $\frac{4}{ac} - \frac{3}{b^{2}} $
• B. $\frac{b^{2}-ac}{a^{2}b^{2}c^{2}} $
• C. $\frac{4}{ac} = \frac{1}{b^{2}} $
• D. None of these

Answer 1

Answer: (A)

$\frac{1}{a}- \frac{1}{b} = \frac{1}{b} - \frac{1}{c} $
$\therefore \left(\frac{1}{a} + \frac{1}{b} - \frac{1}{c}\right)\left(\frac{1}{b} + \frac{1}{c} - \frac{1}{a}\right) $
$=\left(\frac{2}{a} - \frac{1}{b}\right)\left(\frac{2}{c} - \frac{1}{b}\right) = \frac{4}{ac} - \frac{1}{b} \left(\frac{2}{a} + \frac{2}{c}\right) + \frac{1}{b^{2}} $
$ = \frac{4}{ac} - \frac{2}{b}\left(\frac{2}{b}\right) + \frac{1}{b^{2}} = \frac{4}{ac} - \frac{3}{b^{2}} $