Mathematics - Complex Number Question with Solution | TestHub

$|z-(3-2i)|\le 4$ represents a circle whose centre is $(3,-2)$ and radius $=4$. $|z|=|z-0|$ represents the distance of point ' $z$ ' from origin $(0,0)$
Suppose $RS$ is the normal of the circle passing through origin ' $O$ ' and $G$ is its center $(3,-2)$ Here, $OR$ is the least distance and $OS$ is the greatest distance $OR=RG-OG$ and $OS=OG+GS\dots (1)$ As, $RG=GS=4$ $OG=\sqrt{3^2+(-2{)}^2}=\sqrt{9+4}=\sqrt{13}$ From (1), $OR=4-\sqrt{13}$ and $OS=4+\sqrt{13}$ So, required difference $$\begin{array}{rl}& =(4+\sqrt{13})-(4-\sqrt{13})\\ & =\sqrt{13}+\sqrt{13}=2\sqrt{13}\end{array}$$