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If three distinct number $a, b, c$ are in $G.P.$ and the equations $ax^2 + 2bc + c = 0$ and $dx^2 + 2ex + f = 0$ have a common root, then which one of the following statements is correct?
$\frac{d}{a},\frac{e}{b},\frac{f}{c}$ are in $A.P$
$d, e, f$ are in $A.P$
$\frac{d}{a},\frac{e}{b},\frac{f}{c}$ are in $G.P$
$d, e, f$ are in $G.P$
Solution
${b^2} = ac$
Also root of $a{x^2} + 2bx + c = 0$ are equal
$ \Rightarrow x\frac{{ – b}}{a}$
$ \Rightarrow d{\left( {\frac{{ – b}}{a}} \right)^2} + 2e\left( {\frac{{ – b}}{a}} \right) + \int { = 0} $
$d{b^2} – 2aeb + f{a^2} = 0,{b^2} = ac$
$ \Rightarrow dac – 2aeb + f{a^2} = 0$
$ \Rightarrow dc – 7eb + fa = 0$
Dividing by $ac$
$ \Rightarrow \frac{d}{a} – \frac{{2e}}{b} + \frac{f}{c} = 0$
$ \Rightarrow \frac{d}{a} + \frac{f}{c} = 2.\frac{e}{b}$
