Q26 of 66 Page 91

State True of False for the following:

Let z1 and z2 be two complex numbers such that |z1 + z2| = |z1| + |z2|, then arg (z1 – z2) = 0.

False

Explanation:


Given |z1 + z2| = |z1| + |z2|


⇒ |z1 + z2|2 = |z1|2 + |z2|2 + 2 |z1| |z2|


⇒ |z1|2 + |z2|2 + 2 Re (z1z̅ 2) = |z1|2 + |z2|2 + 2 |z1| |z2|


⇒ 2 Re (z1 z̅ 2) = 2 |z1| |z2|


⇒ cos (θ1 – θ2) = 1


⇒ θ1 – θ2 = 0


∴ arg (z1) – arg (z2) = 0


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