Question

If $z$ and $\omega$ are two complex numbers such that $\left|z\right|\le 1,\left|\omega \right|\le 1,$and $\left|z-i\omega \right|=\left|z-i\overline{\omega }\right|=2$. Then $z$ is equal to

• 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 $-1$

Explanation for the correct option.

Find the possible value of $z$.

It is given that $\left|z-i\omega \right|=\left|z-i\overline{\omega }\right|=2$.

It can be interpreted that $z$ lies on the perpendicular bisector of the line joining $i\omega$ and $i\overline{\omega }$.

Since $i\omega$ is the mirror image of $i\overline{\omega }$ in the $x$-axis, the locus of $z$ is the $x$-axis.

So, the complex number $z$ is completely real as it lies in the $x$-axis.

So the possible values of $z$ are $1$ or $-1$.

Hence, the correct option is C.