Question

Describe the region given by |z-i|z||-|z+i|z||=0.

(a) real axis

(b) imaginary axis

(c) circle centered at origin

(d) quadrant 2

This question was posed to me by my college professor while I was bunking the class.

My question comes from Regions in the Complex Plane in section Complex Numbers of Engineering Mathematics

Answer 1

Right option is (a) real axis

Best explanation: |z/|z|-i|=|z/|z|+i|, z≠0 ⇒ z/|z| is unimodular complex number and lies on the perpendicular bisector of i and –i ⇒ z/|z|=±1 ⇒ z=±|z| ⇒ z is real.