∣∣∣z−1z+1∣∣∣=1 ⇒|z−1|=|z+1| ⇒|x−1+iy|=|x+1+iy| ⇒√x2−2x+1+y2=√x2+2x+1+y2 ⇒x2−2x+1+y2=x2+2x+1+y2 ⇒x=0, which represents a straight line!
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