First slide

Algebra of complex numbers

Question

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

Easy

A

0
B

1
C

2
D

none of these

Solution

We have, for $z \in C$
$\begin{array}{l} | 2 i | = | z + (2 i - z) | \leq | z | + | 2 i - z | \\ \Rightarrow 2 \leq | z | + | z - 2 i | \end{array}$
Thus, minimum value of $| z | + | z - 2 i |$ is 2 and it is attained when z =i.

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