Question

Let z and $\omega$ are two complex numbers such that $|z|\le 1,|\omega |\le 1$ and $|z+i\omega |=|z-i\overline{\omega }|=2$, then z equals

• A. 1 or i
• B. i or -i
• C. 1 or -1
• D. i or -1

Answer 1

The correct option is C 1 or -1

$|z+\iota \omega |=|z+\overline{\iota \omega }|=2$

$|z+\overline{z_1}|=|z+z_1|=2$

$z$ is bisector of $z_1$ and $\overline{z_1}$ (image of $z$ wrt x-axis).

so, $z$ lies on x-axis.

Maximum value of $z$ is $1$ or $-1$.