Question

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

• A. A.P.
• B. G.P
• C. H.P.
• D. none of these

Answer 1

Answer: (C)

$\frac{1}{b-a}+\frac{1}{b-c}=\frac{1}{a}+\frac{1}{c}$
$\Rightarrow \frac{1}{b-a}-\frac{1}{c}$
$=\frac{1}{a}-\frac{1}{b-c}$
$\Rightarrow \frac{c-b+a}{\left(b-c\right)c}=\frac{b-c-a}{a\left(b-c\right)}$
$\Rightarrow \frac{1}{\left(b-a\right)c}-\frac{1}{a\left(b-c\right)}$
$\Rightarrow ab-ac=-bc+ac$
$\Rightarrow 2ac=ab+bc=b\left(a+c\right)$
$\Rightarrow b=\frac{2ac}{a+c}$
$\Rightarrow a,b,c$ are in $H.P$.