Question

The number of complex numbers $z$ such that $|z - 1| = |z + 1| = |z - i|$ equals

• A. $1$
• B. $2$
• C. $\infty$
• D. $0$

Answer 1

Answer: (A)

$1$

Answer 2

Let $z = x + iy$ $|z - 1| = |z + 1|\quad\Rightarrow Re \,z = 0 \quad\Rightarrow x = 0$ $|z - 1| = |z - i|\quad\Rightarrow x = y$ $|z + 1| = |z - i|\quad\Rightarrow y = -x$ Only $\left(0, 0\right)$ will satisfy all conditions. $\Rightarrow$ Number of complex number $z = 1$