Question

If $\frac{1}{b-a}+\frac{1}{b-c}=\frac{1}{a}+\frac{1}{c}$, then a, b, c are in

• A. AP
• B. GP
• C. HP
• D. None of these

Answer 1

Answer: (C)

HP

$\frac{1}{b-a}+\frac{1}{b-c}=\frac{1}{a}+\frac{1}{c}$
$\Rightarrow \frac{b-c+b-a}{(b-a)(b-c)}=\frac{a+c}{ac}$
$\Rightarrow \frac{2b-a-c}{b^2+ac-ab-cb}=\frac{a+c}{ac}$
$\Rightarrow 2abc-a^{2}c-{ac}^2={ab}^2+a^{2}c-a^{2}b-acb+{cb}^2+{ac}^2-acb-c^{2}b.$
$\Rightarrow \frac{1}{a}+\frac{1}{c}=\frac{2}{b.}$