Question

If $z=\frac{-i(9+i)}{2-i}$.Find the principal argument and $\left|z\right|$.

Answer 1

Given that,

$z=\frac{-i(9-i)}{2-i}$

$=\frac{1-9i}{2-1}\times \frac{2+i}{2+i}$

$=\frac{2+9-18i+i}{2^2+i^2}$

$=\frac{11-17i}{5}$

$=\frac{11}{5}-\frac{17}{5}i$

$|z|=\sqrt{{\left(\frac{11}{5}\right)}^2+{\left(\frac{17}{5}\right)}^2}$

$=\sqrt{\frac{121+289}{25}}=\sqrt{\frac{410}{25}}$

$|z|=\frac{\sqrt{410}}{5}$

$arg\left(z\right)={\tan }^{-1}\left(\frac{-17/5}{11/5}\right)$

$={\tan }^{-1}(-17/11)$

$arg\left(z\right)=-{\tan }^{-1}\left(17/11\right)$.