Question

If $|z-2+i|\le 2,$then find the least value of $\left|z\right|$

Answer 1

Given:$|z-2+i|\le 2$ .......$\left(1\right)$

$|z-2+i|\ge |\left|z\right|-|2-i||$

$\therefore |z-2+i|\ge |\left|z\right|-\sqrt{5}|$ ......$\left(2\right)$

From $\left(1\right)$ and $\left(2\right)$

$|\left|z\right|-\sqrt{5}|\le |z-2+i|\le 2$

$\therefore |\left|z\right|-\sqrt{5}|\le 2$

$\Rightarrow -2\le \left|z\right|-\sqrt{5}\le 2$

$\Rightarrow \sqrt{5}-2\le \left|z\right|\le \sqrt{5}+2$

Hence the least value of $\left|z\right|$ is $\sqrt{5}-2$