First slide

Algebra of complex numbers

Question

If $z \in C$ and $2 z = | z | + i,$ then $z$ equals

Moderate

A

$\frac{\sqrt{3}}{6} + \frac{1}{2} i$
B

$\frac{\sqrt{3}}{6} + \frac{1}{3} i$
C

$\frac{\sqrt{3}}{6} + \frac{1}{4} i$
D

$\frac{\sqrt{3}}{6} + \frac{1}{6} i$

Solution

$\begin{array}{l} | 2 z |^{2} = | z |^{2} + 1 \\ \Rightarrow 3 | z |^{2} = 1 \Rightarrow | z | = 1 / \sqrt{3} \end{array}$
Thus $z = \frac{\sqrt{3}}{6} + \frac{1}{2} i$ ( Verify the options )

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