Download Equation of Circles

Equation of Circles
General Form
( x  a)  ( y  b)  r
2
2
2
where r is the radius and (a, b) is the centre.
Example

A circle of centre (5, −4) and radius 3 has
equation
( x  5)  ( y  (4))  3
2
2
( x  5)  ( y  4)  9
2
2
2
Centre (−1, 2), radius 7
( x  (1))  ( y  2)  7
2
2
( x  1)  ( y  2)  49
2
2
2
Centre (0, 0), radius 1
x  y 1
2
2
Centre (4, −3) and circle touches x-axis

Radius = 3
( x  4)  ( y  3)  9
2
2
Different Form
x  y  2 gx  2 fy  c  0
2
2
To find out what is g, f and c, expand the general form:
( x  a)2  ( y  b)2  r 2
x 2  2ax  a 2  y 2  2 yb  b2  r 2
Rearrange:
x 2  y 2  2ax  2 yb  a 2  b2  r 2  0
g = -a
centre  (a, b)  (−g, −f).
f = -b
c=
a 2  b2  r 2
Radius = a 2  b2  c  g 2  f 2  c
Example

Given equation of circle is ,
find its centre and radius.
x2  y 2  6x  4 y  9  0
x2  y 2  6x  4 y  9  0
From (1):
 (1)
2 g  6  g  3
2 f  4  f  2
c9
Centre is (3, 2) and radius is
(3) 2  (2) 2  9  2