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.
Best Answer
Hint: you haven't used the information that $a,b,c$ are in geometric progression. You can write $b=ar, c=ar^2$ and plug that into your condition, which simplifies it. You can also set $a=1$, which corresponds to dividing the original equation by $a$-if it is zero your equation is just $0=0$ You can plug the expression of each progression in for the second equation
If you continue to solve $x^2+rx+r^2=0$, you find the roots are proportional to $r$-so geometric progression clearly won't work as that says the two ratios are different.