Question

If a, b, c are in G.P. then $\frac{1}{a^2 - b^2} + \frac{1}{b^2}$ is:

• A. $\frac{1}{c^2 - b^2}$
• B. $\frac{1}{b^2 - c^2}$
• C. $\frac{1}{c^2 - a^2}$
• D. $\frac{1}{b^2 - a^2}$

Answer 1

Answer: (B)

As given : a, b, c are in G.P.
$\Rightarrow b^{2 } = ac $
The given expression :
$\frac{1}{a^{2} - b^{2}} + \frac{1}{b^{2}} $
$= \frac{1}{a^{2} -ac} + \frac{1}{ac}$
[using $b^2 = ac $]
$ = \frac{1}{a\left(a-c\right) } + \frac{1}{ac} = \frac{c+a-c}{ac\left(a-c\right)} $
$= \frac{a}{ac\left(a-c\right)} = \frac{1}{ac -c^{2}} = \frac{1}{b^{2} -c^{2}} $ [using $ ac = b^2 $]