Question

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

|2z + 2 – 4i| = 2

Answer 1

|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

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

= `sqrt(1 + 4 - 4)`

= 1

Aliter: 2|(z + 1 – 2i)| = 2

|z – (– 1 + 2i)| = 1

Centre = – 1 + 2i

Radius = 1