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 $H.P$.