The equation

Question:

The equation $|z-i|=|z-1|, i=\sqrt{-1}$, represents:

  1. (1) a circle of radius $\frac{1}{2}$.

  2. (2) the line through the origin with slope 1 .

  3. (3) a circle of radius 1 .

  4. (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 $1 .$

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