If three distinct number a, b, c are in G.P. | vedclass

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Question

${b^2} = ac$

Also root of $a{x^2} + 2bx + c = 0$ are equal 

$ \Rightarrow x\frac{{ - b}}{a}$

$ \Rightarrow d{\left( {\frac{{ - b}}{

Vedclass · Accepted Answer

$\frac{d}{a},\frac{e}{b},\frac{f}{c}$ are in $A.P$

Vedclass · Answer

${b^2} = ac$

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}$

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?

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