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JEE Main 1981 Mathematics question on Complex Numbers and Quadratic Equations

The complex numbers z=x+iyz = x + iy which satisfy the equation ∣z−5iz+5i∣=1\left|\dfrac{z - 5i}{z + 5i}\right| = 1, lie on
  1. A.
    the XX-axis
  2. B.
    the straight line y=5y = 5
  3. C.
    a circle passing through the origin
  4. D.
    None of the above
Show answer and solution

Correct option: A

Solution

|z - 5i| = |z + 5i| means z is equidistant from 5i and -5i. The locus of points equidistant from (0,5) and (0,-5) is the perpendicular bisector of the segment joining them, which is the X-axis (y = 0). So all such complex numbers z = x + iy satisfy y = 0, lying on the X-axis.

[1981]

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