## Minimum value

Example.

For any complex number z, the minimum value of $|z|+|z-2i|$ is:

a) 0

b) 1

c) 2

d) none of these.

Solution.

We have for $z \in C$, $|2i|=|z+(2i-z)|\leq |z|+|2i-z|$

$\Longrightarrow 2 \leq |z|+|z-2i|$

Thus, the required minimum value is 2 and it is attained for any z lying on the segment joining $z=0$ and $z=2i$.