Question

If z is any complex number satisfying $|z-3-2i|\le 2$, then the maximum value of $|2z-6+5i|$ is _______.

Answer 1

$\Rightarrow 2|z-3+\frac{5}{2}i|=2|z3-2i+2i+\frac{5}{2}i|=2|z-3-2i+\frac{9}{4}i|$

we know that

$\left|z_1\right|-\left|z_2\right|\le |z_1+z_2|\le \left|z_1\right|+\left|z_2\right||z-3-2i+\frac{9}{4}i|=||z-3-2i|-\left|\frac{9}{4}i\right||=|2-\frac{9}{4}|=\frac{5}{2}2|z-3-2i+\frac{9}{4}i|=|2z-6+5i|=2\times \frac{5}{2}=5$

Maximum value of \left| 2z-6+5i \right| =5$