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Question
If z1, z2 and z3, z4 are two pairs of conjugate complex numbers, then find arg`(z_1/z_4)` + arg`(z_2/z_3)`.
Solution
Let the polar form of z1 = r1(cosθ1 + isinθ1)
∴ z2 = `barz_1`
= r1(cosθ1 + isinθ1)
= r1[cos(–θ1) + isin(–θ1)]
Similarly, z3 = r2(cosθ2 + isinθ2)
∴ z4 = `barz_3`
= r2(cosθ2 + isinθ2)
= r2[cos(–θ2) + isin(–θ2)]
arg`(z_1/z_4)` + arg`(z_2/z_3)` = arg(z1) – arg(z4) + arg(z2) – arg(z3)
= θ1 – (–θ2) + (–θ1) – θ2
= θ1 + θ2 – θ1 – θ2
= 0
Hence, arg`(z_1/z_4)` + arg`(z_2/z_3)` = 0.
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