For a complex number z = x + iy , where x, y R , denote z = y + ix . The locus of z satisfying |z + z | = |z - z | in the complex plane is

Step-by-step solution

Given z = x + iy and z = y + ix . z + z = (x + y) + i(x + y) |z + z | = (x + y)^2 + (x + y)^2 = 2 |x + y| z - z = (x - y) + i(y - x) |z - z | = (x - y)^2 + (y - x)^2 = 2 |x - y| The given condition is |z + z | = |z - z | . Substituting the magnitudes: 2 |x + y| = 2 |x - y| Squaring both sides: (x + y)^2 = (x - y)^2 x^2 + y^2 + 2xy = x^2 + y^2 - 2xy 4xy = 0 xy = 0 This implies x = 0 or y = 0 . x = 0 represents the imaginary axis and y = 0 represents the real axis. Thus, the locus of z is the union of the real axis a