Modulus Of Complex Numbers

The modulus |z - a| represents the distance of the point z from the point a in the Argand plane, a geometric viewpoint central to JEE Advanced complex-number problems. The equation |z - 3| = |z + 3| states that z is equidistant from the points 3 and -3 on the real axis. The set of points equidistant from two fixed points is the perpendicular bisector of the segment joining them. Since 3 and -3 are symmetric about the origin on the real axis, their perpendicular bisector is the imaginary axis, the line x = 0. Writing z = x + iy confirms this: (x-3)^2 + y^2 = (x+3)^2 + y^2 gives -6x = 6x, so x = 0. Option real axis would require equidistance from points on the imaginary axis. Option y = x has no symmetry justification here. Option circle of radius 3 confuses distance equality with a fixed distance. Hence z lies on the imaginary axis. Plausibility check: any purely imaginary z = iy is clearly equidistant from 3 and -3 by symmetry, confirming the locus.