(i) Let z = x + iy 

|z – 2 – i| = 3 

⇒ |x + iy – 2 – i| = 3 

⇒ |(x – 2) + i(y – 1)| = 3

= √((x - 2)² + (y -1)²) = 3

Squaring on both sides

(x – 2)² + (y – 1)² = 9 

⇒ x² – 4x + 4 + y² – 2y + 1 – 9 = 0 

⇒ x² + y² – 4x – 2y – 4 = 0 represents a circle 

2g = -4 ⇒ g = -2

2f = -2 ⇒ f = -1 

c = -4 

(a) Centre (-g, -f) = (2, 1) = 2 + i (b) 

(b) Radius = √(g² + f² - c) = √(4 + 1 + 4) = 3

Aliter: |z – (2 + i)| = 3 

Centre = 2 + i 

Radius = 3

(ii) |2(x + iy) + 2 – 4i| = 2 

⇒ |2x + i2y + 2 – 4i| = 2 

⇒ |(2x + 2) + i(2y – 4)| = 2 

⇒ |2(x + 1) + 2i(y – 2)| = 2 

⇒ |(x + 1) + i(y – 2)| = 1

= √((x + 1)² + (y - 2)²) = 1

Squaring on both sides, 

x² + 2x + 1 + y² + 4 – 4y – 1 = 0

⇒ x² + y² + 2x – 4y + 4 = 0 represents a circle 

2g = 2 ⇒ g = 1 

2f = -4 ⇒ f = -2 

c = 4 

(a) Centre (-g, -f) = (-1, 2) = -1 + 2i 

(b) Radius = √(g² + f² - c) = √(1 + 4 - 4) = 1

Aliter: 2|(z + 1 – 2i)| 

= 2 |z – (-1 + 2i)| = 1 

Centre = -1 + 2i 

Radius = 1

(iii) |3(x + iy) – 6 + 12i| = 8 

⇒ |3x + i3y – 6 + 12i| = 8 

⇒ |3(x – 2) + i3 (y + 4)| = 8 

⇒ 3|(x – 2) + i (y + 4)| = 8

= 3√((x - 2)² + (y + 4)²) = 8

Squaring on both sides, 9[(x – 2)² + (y + 4)²] = 64 

⇒ x² – 4x + 4 + y² + 8y + 16 = 64/9

⇒ x² + y² – 4x + 8y + 20 – (64/9) = 0

x² + y² – 4x + 8y + (116/9) = 0 represents a circle. 

2g = -4 ⇒ g = -2 

2f = 8 ⇒ f = 4 

c = 116/9

(a) Centre (-g, -f) = (2, -4) = 2 – 4i 

(b) Radius = √(g² + f² - c) = √(4 + 16 - (116/9)) = √(180 - 116)/9 = 8/3

Aliter: |z – 2 + 4i| = 8/3

⇒ |z – (2 – 4i)| = 8/3

Centre = 2 – 4i, Radius = 8/3