If $a,\;b,\;c$ are in Geometric Progression, then the equations $ax^2+2bx+c=0$ and $dx^2+2ex+f=0$ have a common root if $\;\displaystyle\frac da,\;\frac eb,\;\frac fc$ are in:
- Arithmetic Progression
- Geometric Progression
- Harmonic Progression
Considering the first equation as $a_1x^2+b_1x+c_1=0$ and the second one as $a_2x^2+b_2x+c_2=0$, I applied the condition for the common root of two quadratic equations, i.e, $$(a_1b_2-b_1a_2)(b_1c_2-c_1b_2)=(c_1a_2-a_1c_2)^2$$ However, it gives a large equation in terms of the constants and does not lead me anywhere near finding the relation.