1. “Teach A Level Maths” Vol. 1: AS Core Modules
16: Circles
© Christine Crisp

2. Module C1 Module C2 AQA Edexcel MEI/OCR OCR
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3. P(x, y ) y x Consider a circle, with centre the origin and radius 1
Let P(x, y) be any point on the circle x y O
P(x, y ) 1

4. Consider a circle, with centre the origin and radius 1
Let P(x, y) be any point on the circle
By Pythagoras’ theorem for triangle OPM, x y O
P(x, y ) 1 y M x

5. The equation gives a circle because only the coordinates of points on the circle satisfy the equation.
e.g. Since the radius is 1, we can see that the point (1, 0) lies on the circle x y
(1, 0) x 1

6. = the right hand side (r.h.s.)
The equation gives a circle because only the coordinates of points on the circle satisfy the equation.
e.g. Since the radius is 1, we can see that the point (1, 0) lies on the circle
Substituting (1, 0) in the left hand side (l.h.s.) of the equation
l.h.s.
= the right hand side (r.h.s.)
So, the equation is satisfied by the point (1, 0)

7. y x The point does not lie on the circle since l.h.s. (0. 5, 0. 5) x
r.h.s.
The equation is NOT satisfied by the point (0.5, 0.5).
The point does not lie on the circle.

8. If we have a circle with centre at the origin but with radius r, we can again use Pythagoras’ theorem
P(x, y ) x y O M
We get r

9. Now consider a circle with centre at the point ( a, b ) and radius r.x y
P(x, y )
y - b x
x - a
Using Pythagoras’ theorem as before:

10. Another way of finding the equation of a circle with centre ( a, b ) is to use a translation fromx y x x
Translate by :
Replace x by (x – a) and y by (y – b)

11. SUMMARY
The equation of a circle with centre ( a, b ) and radius r is
We usually leave the equation in this form without multiplying out the brackets

12. Solution: Using the formula,
e.g. Find the equation of the circle with centre ( 4, -3 ) and radius 5. Does the point ( 2, 1 ) lie on, inside, or outside the circle?
Solution: Using the formula,
the circle is
( 4 , -3 )
( 2, 1 ) x
Substituting the coordinates ( 2, 1 ):
l.h.s.
this gives the square of the distance of the point from the centre of the circle
Since the distance of the point from the centre is less than the radius, the point ( 2, 1 ) is inside the circle

13. SUMMARY
The equation of a circle with centre ( a, b ) and radius r is
To determine whether a point lies on, inside, or outside a circle, substitute the coordinates of the point into the l.h.s. of the equation of the circle and compare the answer with

14. Exercises
1. Find the equation of the circle with centre (-1, 2 ) and radius 3. Multiply out the brackets to give your answer in the form
Solution:
Use
a = -1, b = 2, r = 3
2. Determine whether the point (3,-5) lies on, inside or outside the circle with equation
Solution: Substitute x = 3 and y = -5 in l.h.s.
so the point lies outside the circle

15. Finding the centre and radius of a circle
e.g. Find the centre and radius of the circle with equation
Solution:
First complete the square for x

16. Finding the centre and radius of a circle
e.g. Find the centre and radius of the circle with equation
Solution:
First complete the square for x
N.B.
so we need to subtract 9 to get

17. Finding the centre and radius of a circle
e.g. Find the centre and radius of the circle with equation
Solution:
First complete the square for x

18. Finding the centre and radius of a circle
e.g. Find the centre and radius of the circle with equation
Solution:
Next complete the square for y

19. Finding the centre and radius of a circle
e.g. Find the centre and radius of the circle with equation
Solution:
Copy the constant and complete the equation

20. Finding the centre and radius of a circle
e.g. Find the centre and radius of the circle with equation
Solution:
Finally collect the constant terms onto the r.h.s.
we can see the centre is ( 3, 2 ) and the radius is 5.
By comparing with the equation ,

21. SUMMARY
To find the centre and radius of a circle given in a form without brackets:
Complete the square for the x-terms
Complete the square for the y-terms
Collect the constants on the r.h.s.
Compare with
The centre is (a, b) and the radius is r.

22. Exercises
Find the centre and radius of the circle whose equation is (a)
(b)
Solution: Complete the square for x and y:
Centre is ( 2, -4 ) and radius is 4
Solution: Complete the square for x and y:
Centre is and radius is 3

24. The following slides contain repeats of information on earlier slides, shown without colour, so that they can be printed and photocopied.
For most purposes the slides can be printed as “Handouts” with up to 6 slides per sheet.

25. The equation of a circle with centre ( a, b ) and radius r is
We usually leave the equation in this form without multiplying out the brackets
SUMMARY
To determine whether a point lies on, inside, or outside a circle, substitute the coordinates of the point into the l.h.s. of the equation of the circle and compare the answer with

26. Since the distance of the point from the centre is less than the radius, the point ( 2, 1 ) is inside the circle
e.g. Find the equation of the circle with centre ( 4, -3 ) and radius 5. Does the point ( 2, 1 ) lie on, inside, or outside the circle?
Substituting the coordinates ( 2, 1 ):
l.h.s.
Solution: Using the formula,
the circle is

27. To find the centre and radius of a circle given in a form without brackets:
Complete the square for the x-terms
Complete the square for the y-terms
Collect the constants on the r.h.s.
Compare with
The centre is (a, b) and the radius is r.
SUMMARY

28. e.g. Find the centre and radius of the circle with equation
Finally collect the constant terms onto the r.h.s.
Solution:
we can see the centre is ( 3, 2 ) and the radius is 5.
By comparing with the equation ,