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  4. The complex number z satisfying the equation |z-i| = |z+1| = 1 is

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Q. The complex number $z$ satisfying the equation $|z-i| = |z+1| = 1$ is

WBJEEWBJEE 2017Complex Numbers and Quadratic Equations

Solution:

We have, $|z-i|=|z+1|=1$
Let, $ z=x+i y$
$\therefore |z-i|=1$
$\Rightarrow |x+i y-i|=1$
$\Rightarrow |x+(y-1) i|=1$
$\Rightarrow x^{2}+(y-1)^{2}=1\,\,\,\,...(i)$
Also, $|z+1|=1$
$\Rightarrow |x+i y+1|=1$
$\Rightarrow |(x+1)+i y|=1$
$\Rightarrow (x+1)^{2}+y^{2}=1\,\,\,\,...(ii)$
From Eqs. (i) and (ii), we get
$x=y=0$ and $x=-1, y=1$
$\therefore z=0,-1+i$
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