Question

lf $|z-2+2i|=1$, then the least value of $|z|$ is

• A. $\sqrt{8}+1$
• B. $\sqrt{6}+1$
• C. $\sqrt{6}-1$
• D. $\sqrt{8}-1$

Answer 1

The correct option is D $\sqrt{8}-1$

$z-2+2i=z-(2-2i)$
$\therefore |z-(2-2i)|$ represents the distance of ${}^{\prime }{z}^{\prime }$ from ${}^{\prime }2-2i^{\prime }$
$\therefore |z-(2-2i)|=1$ represents a circle with centre as
${}^{\prime }2-2i^{\prime }$ & radius 1.
$|z|$ is least when it lies closest to the origin (given
that it lies on the circle).
$\therefore |z{|}_{min}=|2-2i|-1$
$=\sqrt{2^2+(-2{)}^2}-1$
$=\sqrt{8}-1=(2\sqrt{2}-1)$