Block 1
Circle Tactics
Basic Tactics
If given an equation findCentre Radius
Draw diagrams
Straight line stuff vital
If finding equation needCentre Radius
esp. dist formula
x2 + y2 + 2gx + 2fy + c = 0 (x – a)2 + (y – b)2 = r2
Terminology
Concentric circles?Same centre
CongruentSame shape and size
Diametrically OppositeOpposite ends of diameter
A
B
Circle Tactics
If given an equation findCentre Radius
Draw diagrams
Straight line stuff vital
If finding equation needCentre 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
Terminology
Concentric circles?Same centre
CongruentSame shape and size
Diametrically OppositeOpposite ends of diameter
Using Common Sense!A(1 , 4) and B are diametrically oppositeCentre is (6 , 10) B?
(1,4)
B
(6,10)
use steps
5
65
6
(11,16)
Using Common Sense!A(2 , 6) and B are diametrically oppositeCentre is (5 , 13) B?
(2,6)
B
(5,13)
use steps
3
73
7
(8,20)
ReminderPerpendicular Bisectorcut line in half at right angles x1+x2 y1+y2
2 2
m1m2 = -1
Finding Equation From Points On Circle
If we have three points on a circle A,B and Cthen perpendicular bisectors will meet in thecentre.We can use this to find the equation of acircle
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
Finding Equation From Points On Circle
If we have three points on a circle A,B and Cthen perpendicular bisectors will meet in thecentre.We can use this to find the equation of acircle
Equation from 3 Points
Perpendicular Bisectorsm1m2 = -1 mid pt
Centre – Point of Intersection
RadiusDistance Formula
(y = y or sim. equations)
Closest DistanceFind 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
Closest DistanceClosest distance between 2 circles (given equations)
?
Dist between centres – (sum of radii)
Get centres
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)
AB
6 4
10
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
35
Proving circles touch at one point ExternallyDist between centres = Sum of Radii
Proving Circles Touch
Distance between centres= difference between radii
Internally