Mathematics - Complex Number Question with Solution | TestHub

Given, $|z-3-2i|\le 2$ To find minimum of $|2z-6+5i|$ or $2\left|z-3+\frac{5}{2}i\right|$ Using triangle inequality, i.e. ||$z_1|-|z_2||\le \left|z_1+z_2\right|$ $\therefore \left|z-3+\frac{5}{2}i\right|$ $=\left|z-3-2i+2i+\frac{5}{2}i\right|$ $=\left|(z-3-2i)+\frac{9}{2}i\right|\ge ||z-3-2i\left|-\frac{9}{2}\right|$ $\ge \left|2-\frac{9}{2}\right|\ge \frac{5}{2}\Rightarrow \left|z-3+\frac{5}{2}i\right|\ge \frac{5}{2}$ or $|2z-6+5i|\ge 5$