This document discusses the general equation of a circle and provides examples of finding the center and radius of circles given their equations. The general equation of a circle is given as (x - h)2 + (y - k)2 = r2, where (h,k) represents the center and r is the radius. Three examples are worked through, finding the center and radius for circles with equations x2 + y2 - 4x - 8y + 16 = 0, x2 + y2 - 10x - 20y + 100 = 0, and x2 + y2 - 2x + 1 = 5. Alternative methods for finding the center and radius are also discussed.

1. PRM1016 Mathematics I
Circle
George Tan Geok Shim
Centre for Pre University
Universiti Malaysia Sarawak
This OpenCourseWare@UNIMAS and its related course materials are licensed under
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2. General Equation of a Circle
• The equation of a circle with the centre (h,k) and
radius r units is
• Where g,f, and c are constants, represents a circle.
• Coefficient of and are the same.
• There is no term in
   
2 2 2
2 2 2 2 2
2 2
2 2 0
2 2 0
x h y k r
x y hx ky h k r
x y gx fy c
   
      
    

3. • Completing the squares,
• The centre of circle is
• The radius is
02222
 cfygxyx
    cfgfygx
ffyyggxx
cfygxyx



2222
2222
22
022
022
 ,g f 
cfgr  22

4. Example 1
Find the centre and radius of the following circle.
Solution:
Centre(-2,1) and radius
   
    2012
0151142
015
2
2
2
2
2
2
4
2
4
4
01524
22
22
22
2
22
2
22







 





 














yx
yx
yyxx
yyxx
0152422
 yxyx

5. Example 2
Find the centre and radius of the following circle.
Solution:
Centre(-4,5), Radius = 9 units
04010822
 yxyx
       
8154
40255164
405510448
40
2
10
2
10
10
2
8
2
8
8
22
22
222222
22
2
22
2








 





 













)()(
)()(
yx
yx
yyxx
yyxx

6. Alternative Method
Example 3
Find the centre and radius of the following circle.
Solution:
Compare Radius
Centre(-g,-f)=(-2,1), Radius = 5 units
251222
 xyx
0
02
1
22
022
0242
22
22






f
f
g
g
cfygxyx
xyx
   
5
25
2401
22
22




r
r
cfgr
)(