• question_answer If $a,b,c$ are in G.P., then the equations $a{{x}^{2}}+2bx+c=0$ and $d{{x}^{2}}+2ex+f=0$ have a common root if $\frac{d}{a},\frac{e}{b},\frac{f}{c}$ are in [IIT 1985; Pb. CET 2000; DCE 2000] A) A.P. B) G.P. C) H.P. D) None of these
As given, ${{b}^{2}}=ac$Þ equation $a{{x}^{2}}+2bx+c=0$can be written as $a{{x}^{2}}+2\sqrt{ac}x+c=0$ Þ ${{(\sqrt{a}x+\sqrt{c})}^{2}}=0$ Þ $x=-\sqrt{\frac{c}{a}}$ (repeated root) This must be the common root by hypothesis. So it must satisfy the equation $d{{x}^{2}}+2ex+f=0$ Þ $d\frac{c}{a}-2e\sqrt{\frac{c}{a}}+f=0$ Þ $\frac{d}{a}+\frac{f}{c}=\frac{2e}{c}\sqrt{\frac{c}{a}}=\frac{2e}{b}$ Þ $\frac{d}{a},\frac{e}{b},\frac{f}{c}$are in A.P.