Question

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

|3z – 6 + 12i| = 8 

Answer 1

|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

⇒ `3sqrt((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 + `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 = `sqrt("g"^2 + "f"^2 - "c")`

= `sqrt(4 + 16 - 116/9)`

= `sqrt((180 - 116)/9)`

= `8/3`

Aliter:

|z – 2 + 4i| = `8/3`

⇒ |z – (2 – 4i)| = `8/3`

Centre = 2 – 4i

Radius = `8/3`