# Given two complex numbers $z,w$ such that $|z|=|w|=1$. Find the correct statement.

Mathematics Asked by Broly-29 on December 29, 2020

Given two complex numbers $$z,w$$ with unit modulus (i.e., $$|z|=|w|=1$$), which of the following statements will always be correct?

a.) $$|z+w|ltsqrt2$$ and $$|z-w|ltsqrt2$$

b.) $$|z+w|lesqrt2$$ and $$|z-w|gesqrt2$$

c.) $$|z+w|gesqrt2$$ or $$|z-w|gesqrt2$$

d.) $$|z+w|ltsqrt2$$ or $$|z-w|ltsqrt2$$

It is a multiple choice question and only one option is correct.

My approach: As modulus of $$z$$ and $$w$$ is 1. Let $$z=e^{ialpha_1}$$ and $$w=e^{ialpha_2}$$. Now,

$$|z+w|= |e^{ialpha_1}+e^{ialpha_2}|$$

$$|z+w|=|2 cos(frac{alpha_1-alpha_2}{2})e^{frac{i(alpha_1+alpha_2)}{2}}|$$

$$|z+w|=2 |cos(frac{alpha_1-alpha_2}{2})|$$

I don’t know how to proceed after this. Any help would be appreciated.

## One Answer

(C) holds as a consequence of the parallelogram law:

$$|z+w|^2 + |z-w|^2 = 2(|z|^2+|w|^2) = 4$$ so that at least one of $$|z+w|^2$$ or $$|z-w|^2$$ must be $$ge 2$$.

I leave it to you to find counterexamples for (A), (B), and (D). These can easily be found by choosing $$z, w$$ from $$-1, 1, i$$.

Correct answer by Martin R on December 29, 2020

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