Question

If $\left|z-\left(3+4i\right)\right|\le 3$, then find the complex number having least magnitude satisfying the about inequality.

Answer 1

$|z-(3+4i)|\le 3$

$\Rightarrow z-(3+4i)\le 3$ or $z-(3+4i)\ge -3$

$z\le 3+3+4i$ or $z\ge -3+3+4i$

$z\le 6+4i$ or $z\ge 4i$

Minimum value of $z\Rightarrow \sqrt{(4i{)}^2}=4$

$|z|=-4$.