Circle Tactics

1. Block 1
Circle Tactics

2. Basic Tactics
If given an equation find
Centre
Radius
Draw diagrams
Straight line stuff vital
If finding equation need
Centre
Radius
esp. dist formula
x2
+ y2
+ 2gx + 2fy + c = 0
(x – a)2
+ (y – b)2
= r2

3. Terminology
Concentric circles?
Same centre
Congruent
Same shape and size
Diametrically Opposite
Opposite ends of diameter
A
B

4. Circle Tactics
If given an equation find
Centre
Radius
Draw diagrams
Straight line stuff vital
If finding equation need
Centre
Radius
using x2
+ y2
+ 2gx + 2fy + c = 0
(unless equation given with brackets)
then use (x – a)2
+ (y – b)2
= r2
- especially distance formula

5. Terminology
Concentric circles?
Same centre
Congruent
Same shape and size
Diametrically Opposite
Opposite ends of diameter

6. Using Common Sense!
A(1 , 4) and B are diametrically opposite
Centre is (6 , 10) B?
(1,4)
B
(6,10)
use
steps
5
6
5
6
(11,16)

7. Using Common Sense!
A(2 , 6) and B are diametrically opposite
Centre is (5 , 13) B?
(2,6)
B
(5,13)
use
steps
3
7
3
7
(8,20)

8. Reminder
Perpendicular Bisector
cut line in half at
right angles x1+x2 y1+y2
2 2
m1m2 = -1

9. Finding Equation From Points On Circle
If we have three points on a circle A,B and C
then perpendicular bisectors will meet in the
centre.
We can use this to find the equation of a
circle

10. A (0 , 2) B(1 , 5) and C(4 , 4) lie on a circle.
Find its equation.
Radius
A
B
C
(0 , 2)
(1 , 5)
(4 , 4)
(½, 7
/2)
m1 = 3 m2 = -1
/3
3y = -x + 11
(5
/2 , 9
/2)
m1 = -1
/3
m2 = 3
y = 3x – 3 Lines meet at (2 , 3)
(2 , 3)
√5 (x – 2)2
+ (y – 3)2
= 5

11. Finding Equation From Points On Circle
If we have three points on a circle A,B and C
then perpendicular bisectors will meet in the
centre.
We can use this to find the equation of a
circle

12. Equation from 3 Points
Perpendicular Bisectors
m1m2 = -1 mid pt
Centre – Point
of Intersection
Radius
Distance Formula
(y = y or
sim. equations)

13. Closest Distance
Find the closest distance between
circle A x2
+ y2
– 6x – 2y + 9 = 0
and B x2
+ y2
– 14x – 8y + 61 = 0
(3 , 1)
(7 , 4)
1
2
2
A
B
5

14. Closest Distance
Closest distance between 2 circles
(given equations)
?
Dist between centres – (sum of radii)
Get centres

15. Proving Circles Touch Externally
Prove these touch externally
A x2
+ y2
– 6x – 4y – 23 =
0 B x2
+ y2
– 18x – 20y +
165 = 0
(3 , 2) (9 , 10)
A
B
6
4
10

16. Proving Circles Touch Internally
Prove these touch internally
A x2
+ y2
– 4x – 6y – 51 =
0 B x2
+ y2
– 12x – 12y + 63
= 0
(2 , 3)
(6 , 6) 8
3
5

17. Proving circles touch at one point
Externally
Dist between centres = Sum of Radii

18. Proving Circles Touch
Distance between centres
= difference between radii
Internally