Question

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

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

Answer 1

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

As we know that $|z-z_0|=c$ represents equation of circle.

Here,

$z_0=3-2i$

$c=4$

Maximum value of $z=c+\left|z\right|$

Minimum value of $z=c-\left|z\right|$

For $|z-3+2i|\le 4$

Maximum value $=4+\sqrt{3^2+{(-2)}^2}=4+\sqrt{13}$

Miniimum value $=4-\sqrt{3^2+{(-2)}^2}=4-\sqrt{13}$

Difference between maximum and minimum value

$=(4+\sqrt{13})-(4-\sqrt{13})=4+\sqrt{13}-4+\sqrt{13}=2\sqrt{13}$

Hence, this is the answer.