First slide

Algebra of complex numbers

Question

If $| z | = z + 3 - 2 i,$ then $z$ equals

Moderate

A

$7 / 6 + i$
B

$- 7 / 6 + 2 i$
C

$- 5 / 6 + 2 i$
D

$5 / 6 + 2 i$

Solution

$\begin{array}{l} z = | z | - 3 + 2 i \\ | z |^{2} = (| z | - 3)^{2} + 4 \\ = | z |^{2} - 6 | z | + 9 + 4 \\ \Rightarrow | z | = 13 / 6 \end{array}$
Thus, $z = - 5 / 6 + 2 i$

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