Question

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

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

Answer 1

The correct option is C $8$

Given, $|z-3+2i|\le 4$

$\Rightarrow |z-(3-2i)\le 4$ ………..$\left(1\right)$

Now, $|z|=|z-(3-3i)+(3-2i)|$

$\le |z-(3-2i)|+|3-2i|$

$\le 4+\sqrt{\left(9\right)+(-2{)}^2}$ [from $1$]

$=4+\sqrt{13}$

$\therefore |z|\le 4+\sqrt{13}$

Greatest value of $|z|,g=4+\sqrt{3}$

and $|z|=|z-(3-2i)+(3-2i)|$

$=|(3-2i)+z-(3-2i)|$

$\ge |13-2i)-|z-(3-2i)|$

$\ge |\sqrt{13}-|z-(3-2i)|$

$=\sqrt{13}-|z-(3-2i)|$

$\ge \sqrt{13}-4$ [from $1$]

$\therefore |z|\ge \sqrt{13}-4$

Lowest value of $|z|,l=\sqrt{13}-4$

Difference between the greatest value and the least value of $|z|$

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

$=4+\sqrt{13}-\sqrt{13}+4$

$=8$.