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)]
$\Rightarrow \arg \left(\frac{z_{1}}{z_{4}}\right)+\arg \left(\frac{z_{2}}{z_{3}}\right)=\arg \left(z_{1}\right)-\arg \left(z_{4}\right)+\arg \left(z_{2}\right)-\arg \left(z_{3}\right)$
= θ1 – (-θ2) + (-θ1) – θ2
= θ1 + θ2 – θ1 – θ2
= 0
⇒ arg(z1/z4) + arg(z2/z3) = 0