First slide

Geometry of complex numbers

Question

$The solution of the equation 2 z = | z | + 2 i, where z is a$ $complex number, is -$

Moderate

A

$z = \frac{\sqrt{3}}{3} - i$
B

$z = \frac{\sqrt{3}}{3} + i$
C

$z = \frac{\sqrt{3}}{3} \pm i$
D

None of these

Solution

$\begin{array}{l} 2 (x + i y) = \sqrt{x^{2} + y^{2}} + 2 i \\ 2 x = \sqrt{x^{2} + y^{2}} and 2 y = 2 \Rightarrow y = 1 \\ 4 x^{2} = x^{2} + 1 \Rightarrow 3 x^{2} = 1 \\ i.e., x = \pm \frac{1}{\sqrt{3}} \\ x = \frac{1}{\sqrt{3}} (∵ x \geq 0) \\ ∴ z = \frac{1}{\sqrt{3}} + i = \frac{\sqrt{3}}{3} + i \end{array}$

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