Question: The equation $|z-i|=|z-1|, i=\sqrt{-1}$, represents:
(1) a circle of radius $\frac{1}{2}$.
(2) the line through the origin with slope 1 .
(3) a circle of radius 1 .
(4) the line through the origin with slope $-1$.
Correct Option: , 2
Solution:
Given equation is, $|z-1|=|z-i|$
$\Rightarrow(x-1)^{2}+y^{2}=x^{2}+(y-1)^{2}$$[$ Here, $z=x+i y]$
$\Rightarrow 1-2 x=1-2 y \Rightarrow x-y=0$
Hence, locus is straight line with slope $1 .$
