Mathematics - Complex Number Question with Solution | TestHub

$\begin{array}{rl}& {\mathrm{z}}_1+{\mathrm{z}}_2+{\mathrm{z}}_3=3{\mathrm{z}}_0\\ & {\left({\mathrm{z}}_1+{\mathrm{z}}_2+{\mathrm{z}}_3\right)}^2=9{\mathrm{z}}_0^2\\ & \Rightarrow {\mathrm{z}}_1^2+{\mathrm{z}}_2^2+{\mathrm{z}}_3^2+2\left({\mathrm{z}}_1^2+{\mathrm{z}}_2^2+{\mathrm{z}}_3^2\right)=9{\mathrm{z}}_0^2\\ & \Rightarrow {\mathrm{z}}_1^2+{\mathrm{z}}_2^2+{\mathrm{z}}_3^2=3{\mathrm{z}}_6^2\end{array}$
$\begin{array}{rl}& \sum _{\mathrm{k}=1}^3{\left({\mathrm{z}}_{\mathrm{k}}-{\mathrm{z}}_0\right)}^2={\left({\mathrm{z}}_1-{\mathrm{z}}_0\right)}^2+{\left({\mathrm{z}}_2-{\mathrm{z}}_0\right)}^2+{\left({\mathrm{z}}_3-{\mathrm{z}}_0\right)}^2\\ & ={\mathrm{z}}_1^2+{\mathrm{z}}_2^2+{\mathrm{z}}_3^2+3{\mathrm{z}}_0^2-2\left({\mathrm{z}}_1+{\mathrm{z}}_2+{\mathrm{z}}_3\right){\mathrm{z}}_0\\ & =6{\mathrm{z}}_0^2-6{\mathrm{z}}_0^2\\ & =0\end{array}$