Question

Suppose z and $\omega$ are two complex numbers such that $|z|\le 1,|\omega |\le 1$, and $|z+i\omega |=|z-i\omega |=2$.

Which of the following is true about $|z|$ and $|\omega |$?

• A. $|z|=|\omega |=\frac{1}{2}$
• B. $|z|=\frac{1}{2},|\omega |=\frac{3}{4}$
• C. $|z|=|\omega |=\frac{3}{4}$
• D. $|z|=|\omega |=1$

Answer 1

The correct option is D $|z|=|\omega |=1$

We have,

$|z+i\omega |\le |z|+|i\omega |\le |z|+|i||\omega |\le 2$

$|z-i\omega |\le |z|+|-i\omega |\le |z|+|i||-\omega |=|z|+|i||\omega |\le 2$

Also,

$|z+i\omega |=|z-i\omega |$

and

$|z+i\omega |=2$

Hence,

$|z|=|\omega |=1$