If a,b,c are in G.P. then the equations ${\mathrm{ax}}^2+2\mathrm{bx}+\mathrm{c}=0$ and ${\mathrm{dx}}^2+2\mathrm{ex}+\text{⨍ =0}$ have a common root if d/a , e/b, f/c are in 

$\begin{array}{l}Givena,b,careinG.P\\ \Rightarrow b^2=ac\\ \therefore Discri\min antofa{x}^2+2bx+c=0is\\ D=4b^2-4ac\Rightarrow 4ac-4ac=0\\ \therefore TherootsareEqual\\ Ifa{x}^2+2bx+c=0andd{x}^2+2cx+f=0\\ havecommonrootswhichmeans\\ bothrootsoftheEquationsareequal\\ \therefore \frac{d}{a}=\frac{2e}{2b}=\frac{f}{c}\\ Ifa,b,careinG.Pthend,e,farealsoinG.P\\ \therefore \frac{d}{a},\frac{e}{b},\frac{f}{c}areinG.P\end{array}$