Question

If |z + 2i| = |z – 2i|, then the locus of z is ____________.

Answer 1

Given |z + 2i| = |z – 2i|
for z = |x + iy|
|x + iy + 2i| = |x + iy – 2i|
Squaring both sides, |x + i(y + 2)|² = |x + i (y – 2)|²
i.e x² + (y + 2)|² = x² + (y – 2)|²
i.e x² + y² + 4 + 4y = x² + y² + 4 – 4y
i.e 8y = 0
i.e y = 0
∴ locus is perpendicular bisector of the segment joining (0, –2) and (0, 2)