Question

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

(i) |z – 4| = 16 

(ii) |z – 4|² – |z – 1|² = 16

Answer 1

(i) z = x + iy 

|z – 4| = 16

⇒ |x + iy – 4| = 16 

⇒ |(x – 4) + iy| = 16

= √((x - 4)² + y²) = 16

Squaring on both sides 

(x – 4)² + y² = 256 

⇒ x² – 8x + 16 + y² – 256 = 0 

⇒ x² + y² – 8x – 240 = 0 represents the equation of circle

(ii) |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