Question

Find the complex number $z$, the least in absolute value which satisfies the condition $|z-2+2i|=1.$

Answer 1

The given equation represents points on a circle whose centre
is $C(2,-2)$ in $4th$ quadrant and radius $1$ where $OC$
is inclined at an angle of ${45}^o$ to $x-$ axis. Clearly
the point $P$ on $OC$ will have least absolute value say $r$.
$r=OP=OC-PC=\sqrt{4+4}-1=2\sqrt{2}-1$
$\therefore$ Point $P$ is $(rcos{45}^o-rsin{45}^o)$
$=(\frac{r}{\sqrt{2}},-\frac{r}{\sqrt{2}})=\frac{r}{\sqrt{2}}(1-i)$

$\therefore$ $z=(2-\frac{1}{\sqrt{2}})(1-i).$