If three distinct numbers a, b, c are in G.P. and the equations a x...

∵ a, b, c$are in G.P.$⇒ {b}^{2} = ac \ldots ( i )$<br />Equation$a{x}^{2} + 2bx + c = 0$<br />Hence,$D = 4{b}^{2} - 4ac = 0 \left(∵ {b}^{2} = ac\right)
$⇒$ equation has equal roots $⇒$ roots are =\dfrac{- b}{a}$,$\dfrac{- b}{a} ( $∵$ sum of roots $\alpha + \alpha =$\dfrac{- 2b}{a} )
$∵$ Both equations have one root in common.
$∴$ Common root is $-$\dfrac{b}{a}. It will satisfy equation.
$∴ d{\left(\dfrac{- b}{a}\right)}^{2} + 2e \left(- b\right)a + f = 0 ⇒ d{b}^{2} - 2aeb + {a}^{2}f = 0$
$⇒ d \left(ac\right) - 2aeb + {a}^{2}f = 0 \left(∵ {b}^{2} = ac\right)<br />$⇒ dc - 2aeb + af = 0$<br />$⇒ $\dfrac{dc}{ac} - \dfrac{2eb}{ac} + \dfrac{af}{ac} =0$ (dividing by $ac$ )
$⇒ \dfrac{d}{a} -\dfrac{2eb}{{b}^{2}} + \dfrac{f}{c} =$- ⇒ $\dfrac{d}{a} + \dfrac{f}{c} = \dfrac{2e}{b}$
$⇒ \dfrac{d}{a}, \dfrac{e}{b}, \dfrac{f}{c} a$re in A.P.