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NDA 2016 Maths · Mathematics · Complex Numbers and Quadratic Equations · medium

The locus of the point z in the Argand plane satisfying |z-2|=|z+2i| is

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The locus of the point zz in the Argand plane satisfying ∣z−2∣=∣z+2i∣|z-2|=|z+2i| is
  1. A. the line x−y=0x-y=0
  2. B. a circle with centre at the origin
  3. C. the line x+y=0x+y=0
  4. D. a parabola
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Answer: (C) the line x+y=0x+y=0

The condition says zz is equidistant from 22 and −2i-2i, so it lies on the perpendicular bisector of the segment joining (2,0)(2,0) and (0,−2)(0,-2). Squaring, (x−2)2+y2=x2+(y+2)2(x-2)^2+y^2=x^2+(y+2)^2 gives −4x+4=4y+4-4x+4=4y+4, i.e. x+y=0x+y=0. The bisector passes through the origin with slope −1-1, not +1+1.
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Original question written in the NDA 2016 Maths pattern and checked twice for the correct answer. Not an official question paper. How we write and check questions · Corrections policy · Report a mistake