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If $|z_1| =|z_2|$ and $\arg(z_1/z_2)=π $, then $z_1+z_2$ is

My attempt – We can represent $z_1$ and $z_2$ as two complex numbers at the same distances from origin in the Argand plane but I can't seem to represent the argument in the Argand plane. Can someone help me for this?

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2 Answers 2

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Hint:

The argument of a non-zero complex number corresponds to the polar angle of its affix in polar coordinates; and the argument of a quotient of complex numbers is the difference of their arguments.

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  • $\begingroup$ arg(z1/z2)=arg z1-arg z2 ...this i know.But i am having comfusion in representing it in argand plane.Will the complex numbers lie diagonally opposite to each other or something like that. $\endgroup$
    – Hydrous Caperilla
    Dec 7, 2017 at 23:04
  • $\begingroup$ What does ,a rotation of π correspond to, geometrically, in your opinion? $\endgroup$
    – Bernard
    Dec 7, 2017 at 23:06
  • $\begingroup$ for me,it's like the number is rotated π clockwise $\endgroup$
    – Hydrous Caperilla
    Dec 7, 2017 at 23:10
  • $\begingroup$ That's right, but there's a still simpler interpretation. $\endgroup$
    – Bernard
    Dec 7, 2017 at 23:12
  • $\begingroup$ can u tell me about it $\endgroup$
    – Hydrous Caperilla
    Dec 7, 2017 at 23:15
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There are multiple ways to solve the problem. One way is to see that $$z_1+z_2 = z_2 \left(1+\frac{z_1}{z_2}\right).$$ Now, $\left|\frac{z_1}{z_2}\right|=1$ and $\arg\left(\frac{z_1}{z_2}\right)=\pi$ implies $\frac{z_1}{z_2}=\cos(\pi)+i \sin(\pi) = ? \implies z_1 + z_2 = ?$

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  • $\begingroup$ i got your method but can u help me representing the arguement (z1/z2) in complex plane.I want to solve this question using geometry $\endgroup$
    – Hydrous Caperilla
    Dec 7, 2017 at 23:21

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