Question

If $a,b,c$ are in G.P. then

• A. $a^2{b}^2{c}^2(\frac{1}{a^3}+\frac{1}{a^3}+\frac{1}{c^3})=a^3+b^3+c^3$
• B. $(a^2+b^2)(b^2+c^2)=(b^2+c^2)(a^2+b^2)$
• C. $a^2{b}^2{c}^2(\frac{1}{a^3}-\frac{1}{a^3}-\frac{1}{c^3})=a^3+b^3+c^3$
• D. None of these

Answer 1

The correct option is A $a^2{b}^2{c}^2(\frac{1}{a^3}+\frac{1}{a^3}+\frac{1}{c^3})=a^3+b^3+c^3$

Let, $a=\frac{A}{R}$, $b=A$, $c=AR$ .

$LHS$

$a^2{b}^2{c}^2(\frac{1}{a^3}+\frac{1}{b^3}+\frac{1}{c^3})$

$={\left(\frac{A}{R}\right)}^2{A}^2{\left(AR\right)}^2({\left(\frac{R}{A}\right)}^3+{\left(\frac{1}{A}\right)}^3+{\left(\frac{1}{AR}\right)}^3)$

$\frac{A^2}{R^2}\times A^2\times A^2{R}^2\times (\frac{R^3}{A^3}+\frac{1}{A^3}+\frac{1}{A^3{R}^3})$

$=A^6\left(\frac{R^6+R^3+1}{A^3{R}^3}\right)$

$=\frac{A^3}{R^3}(1+R^3+R^6)$

$RHS$

$a^3+b^3+c^3$

$={\left(\frac{A}{R}\right)}^3+A^3+{\left(AR\right)}^3$

$=\frac{A^3}{R^3}+A^3+A^3{R}^3$

$=\frac{A^3}{R^3}(1+R^3+R^6)$

$\therefore LHS=RHS$

CORRECT OPTION = \left(A\right)