# If three distinct numbers $a,b,c$  are in $G.P.$ and the equations $ax^{2}+2bx+c=0$ and $dx^{2}+2ex+f=0$ have a common root, then which one of the following statements is correct ?    Option 1) $\frac{d}{a},\frac{e}{b},\frac{f}{c}$  are in A.P. Option 2) $d,e,f$ are in A.P. Option 3) $d,e,f$ are in G.P.   Option 4) $\frac{d}{a},\frac{e}{b},\frac{f}{c}$ are in G.P.

Given $a,b,c$ are in G.P.

$\Rightarrow b^{2}=ac$

$Eq:\: ax^{2}+2bx+c=0$

$D=4b^{2}-4ac=0$

roots are = $-\frac{b}{a},-\frac{b}{a}$

$dx^{2}+2ex+f=0$

root of this eq = $\frac{-b}{a}$

$\Rightarrow d\left ( \frac{-b}{a} \right )^{2}+2e\left ( \frac{-b}{a} \right )+f=0$

$\Rightarrow db^{2}-2aeb+a^{2}f=0$

$\because b^{2}=ac$

$\Rightarrow d(ac)-2aeb+a^{2}f=0$

or    $dc-2eb+af=0$

divide by ac

$\frac{dc}{ac}-\frac{2eb}{ac}+\frac{af}{ac}=0$

$\Rightarrow \frac{d}{a}-\frac{2eb}{b^{2}}+\frac{f}{c}=0\Rightarrow \frac{d}{a}+\frac{f}{c}=\frac{2e}{b}$

$\frac{d}{a},\frac{e}{b},\frac{f}{c}\rightarrow AP$

Option 1)

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

Option 2)

$d,e,f$ are in A.P.

Option 3)

$d,e,f$ are in G.P.

Option 4)

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

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