If | z - 2 | = | z - 1 | , where z is a complex number, then locus of z is a straight line

If |z-2|=|z-1|, where z is a complex number, then locus of z is a straight line
  1.  Parallel to x− axis 
  2.  Parallel to y− axis 
  3.  Parallel to y=x
  4.  Parallel to y=-x

Solution

Given |z-2|=|z-1|

On squaring both sides, we get

z-2z¯-2=z-1z¯-1

⇒zz¯-2z¯-2z+4=zz¯-z¯-z+1

⇒z¯+z=3

Now, putting z=x+iy, we get

x+iy+x-iy=3

⇒x=32

Hence, locus of z is a straight line parallel to y-axis

Asked in: AP EAMCET 2021 (19 Aug Shift 2)

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