First slide

Algebra of complex numbers

Question

If $z_{1}, z_{2}$ are two complex numbers such that $|z_{1}| =$ $|z_{2}| = \sqrt{2}$ and $|z_{1} + z_{2}| = \sqrt{3}$ than $|z_{1} - z_{2}|$ equals;

Moderate

A

$2 \sqrt{2}$
B

$\sqrt{5}$
C

3
D

$2 - \sqrt{2}$

Solution

$\begin{array}{l} {|z_{1} + z_{2}|}^{2} + {|z_{1} - z_{2}|}^{2} = 2 {|z_{1}|}^{2} + 2 {|z_{2}|}^{2} \\ \Rightarrow 3 + {|z_{1} - z_{2}|}^{2} = 2 (2) + 2 (2) \\ \Rightarrow {|z_{1} - z_{2}|}^{2} = 5 \Rightarrow |z_{1} - z_{2}| = \sqrt{5} \end{array}$

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