Question

If $|z-3+2i|\le 4$ then the difference between the greatest value and the least value of $|z|$ is

• A. $2\sqrt{113}$
• B. $2\sqrt{13}$
• C. $8$
• D. $4+\sqrt{13}$

Answer 1

The correct option is B $2\sqrt{13}$

given equation represents the circle with center$(3,-2)$ and is of radius $\left(R\right)=4$

$\left|z\right|$ represents the distance of point 'z' from origin

Greatest and least distances occur along the normal through the origin

Normal always passes through center of circle

From figure;

let PQ be the normal through origin 'O'

and C be its center $(3,-2)$

it is clear that OP is the least distance

and OQ is the greatest distance

From diagram;

$OP=CP-OC$ and $OQ=CQ+OC$

Here, $CP=CQ=R=4$

$OC=\sqrt{{(3-0)}^2+{(-2-0)}^2}$

$\Rightarrow OC=\sqrt{13}$

$\therefore OP=CP-OC$

$\Rightarrow OP=4-\sqrt{13}$

$\therefore LeastdistanceOP=4-\sqrt{13}$

$andOQ=CQ+OC$

$\Rightarrow OQ=4+\sqrt{13}$

$\therefore$ Greatest distance $=OQ=4+\sqrt{13}$

Difference between greatest and least distance=OQ-OP=$(4+\sqrt{13})-(4-\sqrt{13})$

$\Rightarrow$ Difference$=2\sqrt{13}$

final answer=$2\sqrt{13}$

the correct option is 'B'