Question

Show that the following equations represent a circle, and, find its centre and radius.

|z – 2 – i| = 3

Answer 1

Let z = x + iy

|z – 2 – i| = 3

⇒ |x + iy – 2 – i| = 3

⇒ |(x – 2) + i(y – 1)| = 3

⇒ `sqrt((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) Radius = `sqrt("g"^2 + "f"^2 - "c")`

= `sqrt(4 + 1 + 4)`

= 3

Aliter: |z – (2 + i)| = 3

Centre = 2 + i

Radius = 3