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

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

AIEEEAIEEE 2010Complex Numbers and Quadratic Equations

Solution:

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 $z = 1$