Question

If $a,b,c$ are distinct non-zero complex numbers such that $|a|=|b|=|c|$.
If $a{z}^2+bz+c=0$ and $b{z}^2+cz+a=0$ have a root of modulus 1 , then the equation $a{z}^2+bz+c=0$ and $b{z}^2+cz+a=0$

• A. have both roots common
• B. have exactly one root common
• C. can not have common root
• D. cannot be determined.

Answer 1

Answer: (A)

$a{z}^2+bz+c=0$ and $b{z}^2+cz+a=0$ have a root of modulus 1
$z_1+z_2=\frac{-b}{a};z_1{z}_2=\frac{c}{a}$
$\left|z_1\right|\left|z_2\right|=1$
$\left|z_2\right|=\left|z_1\right|=1$
$\left|z_1+z_2\right|=1$
$\left(z_1+z_2\right)\left(\frac{1}{z_1}+\frac{1}{z_2}\right)=1$
${\left(z_1+z_2\right)}^2=z_1{z}_2$
$b^2=ac$.....(i)
Similarly for $2^{\text{nd }}$ equation
$c^2=ab$.....(ii)
from (i) and (ii)
$a^3=b^3=c^3$
$\Rightarrow a,b,c$ are $z_0,z_0\omega ,z_0{\omega }^2$.
$\Rightarrow$ ratio of coefficients of two equations are same.
$\Rightarrow$ both roots are same.