First slide

Algebra of complex numbers

Question

If |z₁| = |z₂| and arg (z₁/z₂) = $π$ , then z₁ + z₂ is equal to

Moderate

A

0
B

purely imaginary
C

purely real
D

none of these

Solution

We have
$\arg (\frac{z_{1}}{z_{2}}) = π$
$\begin{aligned} or \arg (z_{1}) - \arg (z_{2}) = π \\ or \arg (z_{1}) = \arg (z_{2}) + π \end{aligned}$
Let $\arg (z_{2}) = θ . Then \arg (z_{1}) = π + θ .$
$\begin{matrix} ∴ z_{1} = |z_{1}| [\cos (π + θ) + isin (π + θ)] \\ = |z_{1}| (- \cos θ - isin θ) \end{matrix}$
$\begin{matrix} and z_{2} = |z_{2}| (\cos θ + isin θ) \\ = |z_{1}| (\cos θ + isin θ) (∵ |z_{1}| = |z_{2}|) \\ = - z_{1} \end{matrix}$
$\Rightarrow z_{1} + z_{2} = 0$

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