Mathematics - Complex Number Question with Solution | TestHub

Let $z=x+iy=r(\cos \theta +i\sin \theta )$, then the equation is

$$\begin{array}{rlr}& |2(x-2)+2i(y-1)|=r\left(\frac{1}{\sqrt{2}}\cos \theta -\frac{1}{\sqrt{2}}\sin \theta \right)=\frac{1}{\sqrt{2}}(r\cos \theta -r\sin \theta )& \Rightarrow \sqrt{(x-2{)}^2+(y-1{)}^2}=\frac{1}{\sqrt{2}}\left(\frac{x-y}{\sqrt{2}}\right)\end{array}$$

It is an ellipse with focus at $(2,1)$ and directrix $x-y=0$ and eccentricity $=\frac{1}{\sqrt{2}}$.