AS Equation of a Circle
Worksheet 1
In small groups – to develop the general equation of a circle.
Matching activity
In small groups – match the equation of a circle to its extended form, centre and radius.
There is one card missing from each set – the students should work these out among
themselves.
You will need the Tarsia programme open to access these.
Circle problems
A range of problems for students to solve in groups.
There are diagrams for them to match to the questions first (they are not in order).
You may want to give them the ‘Useful things to use’ sheet to help.
Further work
The students can design their own exam question.
Equation of a circle (Core 1)
Below is a diagram of a circle, centre (0,0) and radius 5.
Write down an equation connecting x and y for any point on the circle.
What would be the equation of a circle centre (0,0) with radius r?
Using your knowledge of transformations, what is the equation of a circle
centre (3,0), radius 4?
What if the centre was (0,2) and the radius 3?
The general equation of a circle is written when the centre is (p,q) and the
radius r.
What is the general equation of a circle?
Use the Tarsia files to obtain the card matching.
Circle Matching Solutions
(x – 3)2 + (y – 2)2 = 25
x2 + y2 - 6x – 4y - 12 = 0
(3, 2)
5
(x – 3)2 + (y + 2)2 = 16
x2 + y2 - 6x + 4y - 3 = 0
(3, -2)
4
(x – 2)2 + (y + 3)2 = 9
x2 + y2 - 4x + 6y + 4 = 0
(2, -3)
3
(x + 2)2 + (y – 3)2 = 4
x2 + y2 + 4x – 6y + 9 = 0
(-2, 3)
2
(x – 1)2 + (y – 4)2 = 2
x2 + y2 - 2x – 8y + 15 = 0
(1, 4)
√2
(x – 4)2 + (y – 1)2 = 9
x2 + y2 - 8x – 2y + 8 = 0
(4, 1)
3
(x + 1)2 + (y + 2)2 = 16
x2 + y2 + 2x + 4y - 11 = 0
(-1, -2)
4
(x – 2)2 + (y – 1)2 = 25
x2 + y2 - 4x – 2y - 20 = 0
(2, 1)
5
The underlined answers are the missing ones.
Circle problems
Match each question with one of the diagrams on the next page, choose which
steps you need to take to reach a solution and write a full solution.
1. Points A and B have co-ordinates (-3,-6) and (9,2) respectively. Find
the equation of the circle with AB as diameter.
2. Find the equation of the tangent to the circle x2 + y2 + 10x + 2y + 13 =
0 at the point (-3,2).
3. Find the equation of the normal to the circle x2 + (y + 3)2 = 18 at the
point (3,0).
4. The straight line y = 20 – 3x meets the circle x2 + y2 – 2x – 14y = 0 at
the points A and B. Calculate the exact length of the chord AB.
5. Write the equation of the circle centre (0,0), radius 5.
P is the point (3,-4).
Find the points where the tangent to the circle at P crosses the axes
and hence find the area of the triangle formed by these points and the
origin.
6. Three points are P(-2,7), Q(2,3) and R(4,5). Show that PQ is
perpendicular to QR.
Find the equation of the circle which passes through the points P, Q
and R.
Circle Problem Diagrams
Circle Problems
Useful things to do
Find the centre of the circle
Find the radius of the circle
Find the equation of the circle
Multiply out the brackets and re-arrange
Complete the square
Find the midpoint
Find the gradient of the radius
Find the gradient of the tangent (perpendicular to the radius)
Find the length of the radius
Use simultaneous equations
Useful formulae
(x – p)2 + (y – q)2 = r2
x2 + y2 + ax + by + c = 0
M = (½(x1 + x2), ½(y1 + y2))
m = y2 – y1
x2 - x1
L = √((x2 - x1)2 + (y2 – y1)2)
y – y1 = m (x – x1)