Question

$z_1$ and $z_2$ are the roots of the equation $z^2-az+b=0$, where $|z_1|=|z_2|=1$ and a, b are non zero complex numbers, then

• A. $|a|\le 1$
• B. $|a|\le 2$
• C. $arg\left(a^2\right)=arg\left(b\right)$
• D. $arga=arg\left(b^2\right)$

Answer 1

Answer: (B), (C)

$|a|\le 2$
$arg\left(a^2\right)=arg\left(b\right)$

$z_1$ and $z_2$ are the roots of the equation $z^2-az+b=0$. Hence,
$z_1+z_2=a,z_1{z}_2=b$
Now, $|z_1+z_2|\le |z_1|+|z_2|$
$\Rightarrow |z_1+z_1|=|a|\le 1+1=2(\because |z_1|=|z_2|=1)$
$\Rightarrow arg\left(a\right)=\frac{1}{2}\left[arg\right(z_2)+arg(z_1\left)\right]$
Also, $arg\left(b\right)=arg\left(z_1{z}_2\right)=arg\left(z_1\right)+arg\left(z_2\right)$
$\Rightarrow 2arg\left(a\right)=arg\left(b\right)$
$\Rightarrow arg\left(a^2\right)=arg\left(b\right)$