First slide

Algebra of complex numbers

Question

$Let z be a complex number such that |\frac{z - i}{z + 2 i}| = 1 and | z | = \frac{5}{2}, Then the value of |z + 3 i| is:$

Moderate

A

$\sqrt{10}$
B

$2 \sqrt{3}$
C

$\frac{15}{4}$
D

$\frac{7}{2}$

Solution

$\begin{array}{l} Given | z - i |^{2} = | z + 2 i |^{2} \\ \begin{array}{l} \Rightarrow x^{2} + (y - 1)^{2} = x^{2} + (y + 2)^{2} \\ \Rightarrow - 2 y + 1 = 4 y + 4 \\ \Rightarrow y = - \frac{1}{2} \\ Given | z | = \frac{5}{2} \end{array} \end{array}$
It gives
$\begin{matrix} x^{2} + y^{2} = \frac{25}{4} \\ x^{2} = \frac{24}{4} \\ = 6 \\ x = \pm \sqrt{6} \end{matrix}$
$\begin{array}{l} It gives z = \pm \sqrt{6} - \frac{i}{2} \\ The required value is | z + 3 i | = |\sqrt{6} + \frac{5 i}{2}| = \sqrt{6 + \frac{25}{4}} = ∣ \frac{7}{2} \end{array}$

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