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Question
Show that the following equations represent a circle, and, find its centre and radius.
|z – 2 – i| = 3
Solution
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
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