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NDA 2025 Maths · Mathematics · Complex Numbers and Quadratic Equations · medium

If z=x+iy satisfies |(z-1)/(z+1)|=1, then the locus of z is

Single correct option

If z=x+iyz=x+iy satisfies ∣z−1z+1∣=1\left|\dfrac{z-1}{z+1}\right|=1, then the locus of zz is
  1. A. the xx-axis
  2. B. the yy-axis
  3. C. the circle ∣z∣=1|z|=1
  4. D. the line x+y=0x+y=0
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Answer: (B) the yy-axis

The condition means ∣z−1∣=∣z+1∣|z-1|=|z+1|, i.e. (x−1)2+y2=(x+1)2+y2(x-1)^2+y^2=(x+1)^2+y^2, which simplifies to −4x=0-4x=0, so x=0x=0: the yy-axis. Geometrically zz is equidistant from 11 and −1-1, so it lies on their perpendicular bisector.
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