If three distinct numbers a,b,c are in G.P. and the equations $a x^{2}+2 b x+c=0$ and $\mathrm{dx}^{2}+2 \mathrm{ex}+f=0$ have a common root, then which one of the following statements is correct?
Correct Option: , 3
$\mathrm{a}, \mathrm{b}, \mathrm{c}$ in G.P.
say a, ar, ar $^{2}$
satisfies $a x^{2}+2 b x+c=0 \Rightarrow x=-r$
$x=-r$ is the common root, satisfies second equation $\mathrm{d}(-\mathrm{r})^{2}+2 \mathrm{e}(-\mathrm{r})+\mathrm{f}=0$
$\Rightarrow \mathrm{d} \cdot \frac{\mathrm{c}}{\mathrm{a}}-\frac{2 \mathrm{ce}}{\mathrm{b}}+\mathrm{f}=0$
$\Rightarrow \frac{\mathrm{d}}{\mathrm{a}}+\frac{\mathrm{f}}{\mathrm{c}}=\frac{2 \mathrm{e}}{\mathrm{b}}$
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