Question

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

• 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)

$a z^2+b z+c=0$ and $b z^2+c z+a=0$ 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.