First slide

Algebra of complex numbers

Question

Suppose $a \in R$ and the equation $z + a | z | +$ $2 i = 0$ has no solution in $C,$ then $α$ satisfies the relation.

Moderate

A

$| a | > 1$
B

$| a | \geq 1$
C

$| a | > \sqrt{2}$
D

$| a | \geq \sqrt{2}$

Solution

$z = - a | z | - 2 i$
$\begin{array}{lrl} \Rightarrow | z |^{2} = a^{2} | z |^{2} + 4 \\ \Rightarrow | z |^{2} (1 - a^{2}) = 4 \end{array}$
This equation has no solution if $1 - a^{2} \leq 0 or | a | \geq 1$
For $| a | < 1, | z | = \frac{2}{\sqrt{1 - a^{2}}}$ and
$z = \frac{2 a}{\sqrt{1 - a^{2}}} - 2 i$

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