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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:
Hypotenuse
Pythagoras
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 )

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

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

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
Answers
1. 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?

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  2
r
2
)( by 
Using Pythagoras’ theorem as before:

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 circle
If 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 Circle
The 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

A 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

1) 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=6
HOME
Answers
1a) 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 -7.5
b) y=11.3 or -3.3
Answers
2) 16
3)Circumference = 25.1
Area=50.3
4)Circumference = 25.1
Area=50.3
5) Circles have the same
radius but different centres,
they are translations

Worksheet
Answers
1. (0,0) radius 6
2. (2, 7) radius 7
3. (-1, -6) radius 4
4. (-3, 11) radius √12
5. x2 + y2 = 49
6. (x – 4)2 + (y – 3)2 = 64
7. (x – 5)2 + (y – 3)2 = 4
8. (x + 5)2 + (y – 4)2 = 0.25
9. (x + 2)2 + (y + 5)2 = 2
10.(x + 1)2 + (y – 6)2 = 5

Worksheet
Answers
11. x2 + y2 = 4
12. (x + 3)2 + (y -3)2 = 1
13. x2 + (y – 3)2 = 16
14. (x – 7)2 + (y + 2)2 = 4
15. x2 + (y + 20)2 = 100
16. (x + 4)2 + (y + 5)2 = 25

Worksheet
Answers
17. 18.
19. 20.

Worksheet
Answers
21. x2 + y2 = 25
22. (x – 5)2 + (y – 9)2 = 9
23. (x + 5)2 + (y + 9)2 = √61
24. (x – 7)2 + (y + 2)2 = √80
25. (x + 4)2 + (y + 3)2 = 4
26. (x – 4)2 + (y – 1)2 = 16

Now identify
WWW – (what did you do well)
EBI – (what areas do you need to
improve on)

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Equation of a circle

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