Equation of a circle

LO: To derive the equation of a circle, and find the centre and radius.

Underline Title and Date. Put C/W next to title.

Key words:HypotenusePythagoras

Right angled triangle

Equation

Match the words to the definitions

•Sector

•Segment

•Chord

•Radius

•Arc

•Tangent

•Diameter

•Circumference

•The length around the outside of a circle•A line which just touches a circle at one point•A section of a circle which looks like a slice of pizza•A section circle formed with an arc and a chord•The distance from the centre of a circle to the edge•The distance from one side of a circle to the other (through the centre)•A section of the curved surface of a circle•A straight line connecting two points on the edge of a circle

Extension – illustrate these in a diagram

The equation of a circle

x

y

O

1

Consider a circle, with centre the origin and radius 1 Let P(x, y) be any point on the

circle

P(x, y )

The equation of a circle

x

y

O

P(x, y )

1

Consider a circle, with centre the origin and radius 1 Let P(x, y) be any point on the

circle

x

y

By Pythagoras’ theorem for triangle OPM, 122 yx

M

The equation of a circle

P(x, y )

x

y

O x

y

M

P(x, y )

x

y

O x

y

M

If we have a circle with centre at the origin but with radius r, we can again use Pythagoras’ theorem

r

222 ryx

We get

The equation of a circle

So a circle with the centre at 0,0 and a radius of 5 will have the equation x2 + y2 = 25

1. Radius 6

2. Radius 7

3. Radius 9

4. Radius 10

x2 + y2 = 4

Answers1. x2 + y2 = 36

2. x2 + y2 = 49

3. x2 + y2 = 81

4. x2 + y2 = 100

Write the equation of these circle all with a centre at 0,0

What will the equation be for a circle with a centre at 0,0 and a radius of 2?

The equation of a circle

x

y

Now consider a circle with centre at the point ( a, b ) and radius r.

x ),( ba

r

P(x, y )

x - a

y - b

2)( ax 2r2)( by Using Pythagoras’ theorem as before:

The equation of a circle

The equation of a circle with centre ( a, b ) and radius r is

222 )()( rbyax

We usually leave the equation in this form without multiplying out the brackets

SUMMARY

Writing the equation of a circleIf you are given the centre and the radius, you can write the equation of the circle.Example; A circle has the centre 3, -2 and a radius of 3. What is the equation of the circle?

The general equation for a circle is (x-a)2 + (y-b)2=r2

So (x-3)2 + (y+2)2=32

So (x-3)2 + (y+2)2=9

Your turn a circle has a centre 5, -3 and a radius of 8. What is the equation of this circle?

Mini whiteboards at the ready!

The Equation of a CircleThe general equation for a circle is (x-a)2 + (y-b)2=r2

This equation will give a circle whose centre is at (a,b) and has a radius of r

For example a circle has the equation (x-2)2 + (y-3)2=52

This equation will give a circle whose centre is at (2,3) and has a radius of 5

The Equation of a CircleA circle has the equation (x-5)2 + (y-7)2=16

This equation will give a circle whose centre is at (5,7) and has a radius of 4 (square root of 16 is 4)

For example a circle has the equation (x+2)2 + (y-4)2=100

This equation will give a circle whose centre is at (-2,4) and has a radius of 10. http://

www.mathwarehouse.com/geometry/circle/equation-of-a-circle.php

You could think of this as (x - -2)2

The Equation of a Circle1) Write down the coordinates of the centre point and radius of each of these circles:

a) (x-5)2 + (y-7)2=16

b) (x-3)2 + (y-8)2=36

c) (x+2)2 + (y-5)2=100

d) (x+2)2 + (y+5)2=49

e) (x-6)2 + (y+4)2=144

f) x2 + y2=4

g) x2 + (y+4)2=121

h) (x-1)2 + (y+14)2 -16=0

i) (x-5)2 + (y-9)2 -10=15

2) What is the diameter of a circle with the equation (x-1)2 + (y+3)2 =64

3) Calculate the area and circumference of the circle with the equation (x-5)2 + (y-7)2=16

4) Calculate the area and perimeter of the circle with the equation (x-3)2 + (y-5)2=16

5) Compare your answers to question 3 and 4, what do you notice, can you explain this?

6 ) A circle has the equation (x+2)2 + (y-4)2=100, find:

a) x when y=7

b) y when x=6HOME

Answers1a) r=4 centre (5,7)b) r=6 centre (3,8)c) r=4 centre (-2,5)d) r=10 centre (-2,-5)e) r=7 centre (6,-4)f) r=12 centre (0,0)g) r=411centre (0,-4)h) r=4 centre (1,-14)i) r=5 centre (5,9)

6a) x= 11.5 or -7.5b) y=11.3 or -3.3

Answers2) 163)Circumference = 25.1 Area=50.34)Circumference = 25.1 Area=50.35) Circles have the same radius but different centres, they are translations

WorksheetAnswers1. (0,0) radius 62. (2, 7) radius 73. (-1, -6) radius 44. (-3, 11) radius √125. x2 + y2 = 496. (x – 4)2 + (y – 3)2 = 647. (x – 5)2 + (y – 3)2 = 48. (x + 5)2 + (y – 4)2 = 0.259. (x + 2)2 + (y + 5)2 = 210.(x + 1)2 + (y – 6)2 = 5

WorksheetAnswers11. x2 + y2 = 412. (x + 3)2 + (y -3)2 = 113. x2 + (y – 3)2 = 1614. (x – 7)2 + (y + 2)2 = 415. x2 + (y + 20)2 = 10016. (x + 4)2 + (y + 5)2 = 25

WorksheetAnswers17. 18.

19. 20.

WorksheetAnswers21. x2 + y2 = 2522. (x – 5)2 + (y – 9)2 = 923. (x + 5)2 + (y + 9)2 = √6124. (x – 7)2 + (y + 2)2 = √8025. (x + 4)2 + (y + 3)2 = 426. (x – 4)2 + (y – 1)2 = 16

Now identify

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