Mathematics - Complex Number Question with Solution | TestHub

$\begin{array}{rl}& |\mathrm{z}-1|\le 1\\ & \Rightarrow |(\mathrm{x}-1)+\mathrm{iy}|\le 1\mid \\ & \Rightarrow \sqrt{(\mathrm{x}-1{)}^2+{\mathrm{y}}^2}\le 1\\ & \Rightarrow (\mathrm{x}-1{)}^2+{\mathrm{y}}^2\le 1\dots (1)\end{array}$ $\begin{array}{rl}& \text{ Also }|z-5|\le |z-5i|\\ & (x-5{)}^2+y^2\le x^2+(y-5{)}^2\\ & -10x\le -10y\\ & \Rightarrow x\ge y.....(2)\end{array}$ $\begin{array}{rl}& \text{ Solving (1) and (2) }\\ & \begin{array}{l}\Rightarrow (\mathrm{x}-1{)}^2+{\mathrm{x}}^2=1\\ \Rightarrow 2{\mathrm{x}}^2-2\mathrm{x}=0\\ \Rightarrow \mathrm{x}(\mathrm{x}-1)=0\\ \Rightarrow \mathrm{x}=0\text{ or }\mathrm{x}=1\\ \mathrm{y}=0\text{ or }\mathrm{y}=1\end{array}\end{array}$
Given $\mathrm{x},\mathrm{y}\in \mathrm{I}$ Points $(0,0),(1,0),(2,0),(1,1),(1,-1)$ to find $\begin{array}{rl}& {\left|z_1\right|}^2+{\left|z_2\right|}^2+{\left|z_3\right|}^2+{\left|z_4\right|}^2+{\left|z_5\right|}^2\\ & =0+1+4+1+1+1+1=9\end{array}$