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# If $a, b, c$ are in $GP$ and equations $ax^2+2bx+c=0$ and $dx^2+2ex+f=0$ have a common root, then $\large\frac{d}{a}$, $\large\frac{e}{b}$, $\large\frac{f}{c}$ are in

$(a)\;AP\qquad(b)\;GP\qquad(c)\;HP\qquad(d)\;None\;of\;these$

Explanation : a , b , c are in GP
$b^2=ac$
$ax^2+2bx+c=0$
$ax^2+2\sqrt{ac}\;x+c=0$
$(\sqrt{a}+\sqrt{c})^2=0$
$x=-\sqrt{\large\frac{c}{a}}$
This satisfies $dx^2+2ex+f=0$
$d\;(-\sqrt{\large\frac{c}{a}})^2+2e\;(-\sqrt{\large\frac{c}{a}})+f=0$
$\large\frac{dc}{a}-2e\;\sqrt{\large\frac{c}{a}}+f=0$
$(\large\frac{dc}{a}+f)=2e\;\sqrt{\large\frac{c}{a}}$
$\large\frac{d}{a}+\large\frac{f}{c}=\large\frac{2e}{\sqrt{ac}}$
$\large\frac{d}{a}+\large\frac{f}{c}=\large\frac{2e}{b}$
$\large\frac{d}{a}\;,\large\frac{e}{b}\;\large\frac{f}{c}$$\;are\;in$$\;AP\;.$
edited Mar 13, 2014