Question

If all the roots of $z^3+a{z}^2+bz+c=0$ are of unit modulus, then

• A. $|3-4i+b|>8$
• B. $|c|\ge 3$
• C. $|3-4i+a|\le 8$
• D. $|3-4i+a|>8$

Answer 1

The correct option is C $|3-4i+a|\le 8$

Let $\alpha ,\beta ,\gamma$ are roots of the equation
$\therefore \sum \alpha =-a\sum \alpha \beta =b\alpha \beta \gamma =-c$
Now, $|\alpha +\beta +\gamma |\le |\alpha |+|\beta |+|\gamma |\Rightarrow |a|\le 3|\alpha \beta +\beta \gamma +\gamma \alpha |\le |\alpha \beta |+|\beta \gamma |+|\gamma \alpha |\Rightarrow |b|\le 3|\alpha \beta \gamma |=|\alpha ||\beta ||\gamma |=1\Rightarrow |c|=1$
Now, $|3-4i+b|\le |3-4i+b|\le |3-4i|+|b|\le 5+3=8$
Similarly, $|3-4i+q|\le 8$