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If z1, z2 and z3, z4 are two pairs of conjugate complex numbers,

Question:

 If z1, z2 and z3, z4 are two pairs of conjugate complex numbers, then find arg(z1/z4) + arg(z2/z3).

Solution:

According to the question,

We have,

z1 and z2 are conjugate complex numbers.

The negative side of the real axis

= r1 (cos θ1 – i sin θ1)

= r1 [cos (-θ1) + I sin (-θ1)]

Similarly, z3 = r2 (cos θ2 – i sin θ2)

⇒ z4 = r2 [cos (-θ2) + I sin (-θ2)]

$\Rightarrow \arg \left(\frac{\mathrm{z}_{1}}{\mathrm{z}_{4}}\right)+\arg \left(\frac{\mathrm{z}_{2}}{\mathrm{z}_{3}}\right)=\arg \left(\mathrm{z}_{1}\right)-\arg \left(\mathrm{z}_{4}\right)+\arg \left(\mathrm{z}_{2}\right)-\arg \left(\mathrm{z}_{3}\right)$

= θ1 – (-θ2) + (-θ1) – θ2

= θ1 + θ2 – θ1 – θ2

= 0

⇒ arg(z1/z4) + arg(z2/z3) = 0

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