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