First slide

Algebra of complex numbers

Question

$z$ and $ω$ are two non zero complex numbers such that $| z | = | ω |$ and $A r g z + A r g ω = π$ then $z$ equals

Easy

A

$\bar{ω}$
B

$- \bar{ω}$
C

$ω$
D

$- ω$

Solution

Let $| z | = | ω | = r$
$∴ z = r e^{i θ}, ω = r e^{i ϕ}$ , where $θ + ϕ = π (∵ \bar{ω} = r e^{- i ϕ})$

$∴ z = r e^{i (π - ϕ)} = r e^{i π} . e^{- i ϕ} = - r e^{- i ϕ} = - \bar{ω}$

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