Question

Obtain the Cartesian equation for the locus of z = x + iy in the following cases:

|z – 4|² – |z – 1|² = 16

Answer 1

|x + iy – 4|² – |x + iy – 1|² = 16

⇒ |(x – 4) + iy|² – |(x – 1) + iy|² = 16

⇒ [(x – 4)² + y²] – [(x – 1)² + y²] = 16

⇒ (x² – 8x + 16 + y²) – (x² – 2x + 1 + y²) = 16

⇒ x² + y² – 8x + 16 – x² + 2x – 1 – y² = 16

⇒ – 6x + 15 = 16

⇒ 6x + 1 = 0

The locus of the point is a straight line.