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
Correct Answer: A
Solution :
[a] \[\frac{1}{a}-\frac{1}{b}=\frac{1}{b}-\frac{1}{c}\] \[\therefore \,\,\,\,\,\,\left( \frac{1}{a}+\frac{1}{b}-\frac{1}{c} \right)\left( \frac{1}{b}+\frac{1}{c}-\frac{1}{a} \right)\] \[=\left( \frac{2}{a}-\frac{1}{b} \right)\left( \frac{2}{c}-\frac{1}{b} \right)=\frac{4}{ac}-\frac{1}{b}\left( \frac{2}{a}+\frac{2}{c} \right)+\frac{1}{{{b}^{2}}}\] \[=\frac{4}{ac}-\frac{2}{b}\left( \frac{2}{b} \right)+\frac{1}{{{b}^{2}}}=\frac{4}{ac}-\frac{3}{{{b}^{2}}}\]
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