AQA Level 2 Further Maths · Level 2 Certificate · Chapter 5: Coordinate geometry · Lesson 4

The equation of a circle

About 80 minutesNo calculator69 marks

Hints online, answers free with an account:
mathswariz.uk/worksheets/further-maths-the-equation-of-a-circle/

Date

Do Now

The first questions are what today's lesson needs. The others bring back earlier topics.

A1
Needed today
Work out the distance between the points $(- 2, 5)$ and $(4, - 3)$ .
[2 marks]
Answer
A2
Needed today
Write $x^{2} - 10 x + 7$ in the form $(x - p)^{2} - q$ .
[2 marks]
Answer $\dots\dots\dots\dots\dots\dots$
A3
Last lesson
$A$ is $(- 4, 1)$ and $B$ is $(8, 10)$ . $C$ is the point on $A B$ with $A C : C B = 2 : 1$ . Work out the coordinates of $C$ .
[2 marks]
Answer $(\dots\dots, \dots\dots)$
A4
From chapter 1
Write $80$ in the form $a b$ , where $b$ is as small as possible.
[2 marks]
Answer

Example, then your turn

Level 2

Your teacher works through each example with you. Then try the one beside it.

Example 1

(a) Write down the equation of the circle with centre $(3, - 4)$ and radius $6$ . (b) Write down the centre and the radius of the circle $x^{2} + (y + 2)^{2} = 49$ .

Hint A circle with centre

(a, b)

and radius

r

(x - a)^{2} + (y - b)^{2} = r^{2}

. Watch the signs.

B1
(a) Write down the equation of the circle with centre $(- 5, 2)$ and radius $32$ . (b) Write down the centre and the radius of the circle $(x - 6)^{2} + y^{2} = 28$ . Give the radius as a surd in its simplest form.
[3 marks]
Answer(a) $\dots\dots\dots\dots$ (b) $(\dots, \dots)$ , $r = \dots$

Example 2

A circle has centre $C (2, - 1)$ and passes through the point $P (8, 7)$ . Work out the equation of the circle.

Not drawn accurately

Hint The radius is the distance

C P

B2
A circle has centre $(- 3, 5)$ and passes through the point $(2, 17)$ . Work out the equation of the circle.
[3 marks]
Answer $\dots\dots\dots\dots\dots\dots$

Example 3

$A (- 5, 2)$ and $B (3, - 4)$ are the ends of a diameter of a circle. Work out the equation of the circle.

Not drawn accurately

Hint The centre is the midpoint of

A B

. The radius is half of

A B

, or the distance from the centre to

A

B3
$P (- 6, - 3)$ and $Q (4, 7)$ are the ends of a diameter of a circle. Work out the equation of the circle.
[4 marks]
Answer $\dots\dots\dots\dots\dots\dots$

Example 4

Show that $x^{2} + y^{2} + 6 x - 4 y - 12 = 0$ is the equation of a circle. Work out its centre and its radius.

Hint Complete the square in

x

and in

y

separately.

B4
A circle has equation $x^{2} + y^{2} - 12 x + 4 y + 4 = 0$ . Work out the coordinates of its centre and its radius.
[3 marks]
Answercentre $(\dots, \dots)$ radius $\dots\dots$

Example 5

A circle has equation $(x - 2)^{2} + (y + 1)^{2} = 40$ . Is each point inside, on or outside the circle? $A (8, 1)$ , $B (- 3, 4)$ , $C (4, 4)$

Hint Put each point into the left-hand side. Equal to

40

: on. Less than

40

: inside. More than

40

: outside.

B5
A circle has equation $(x + 2)^{2} + (y - 1)^{2} = 34$ . Is each point inside, on or outside the circle? Show your working. $P (3, 4)$ , $Q (1, - 5)$ , $R (- 5, 4)$
[3 marks]
Answer $P$ $\dots\dots$ $Q$ $\dots\dots$ $R$ $\dots\dots$

Practice

Level 2

C1
A circle has centre $(5, - 2)$ . The area of the circle is $45 π$ . Work out the equation of the circle.
[2 marks]
Answer $\dots\dots\dots\dots\dots\dots$
C2
A circle of radius $10$ passes through the points $(- 1, 0)$ and $(11, 0)$ . There are two possible circles. Work out the equation of each one.
[4 marks]
Answer $\dots\dots\dots\dots$ and $\dots\dots\dots\dots$

C3
Show that $x^{2} + y^{2} - 6 x + 10 y + 40 = 0$ is not the equation of a circle.
[2 marks]
C4
The point $(a, 2)$ lies on the circle $(x - 1)^{2} + (y + 3)^{2} = 50$ . Work out the two possible values of $a$ .
[3 marks]
Answer $a = \dots\dots$ or $a = \dots\dots$
C5
Work out the coordinates of every point where the circle $(x - 4)^{2} + (y + 1)^{2} = 17$ meets the $x$ -axis or the $y$ -axis.
[4 marks]
Answer $\dots\dots\dots\dots\dots\dots$

Exam-style questions

AQA exam style

D1
Circle the radius of the circle $x^{2} + (y + 3)^{2} = 18$ . $189323$
[1 mark]
Answer

D2
Sam says the circle $x^{2} + y^{2} - 8 x + 2 y = 8$ has centre $(- 4, 1)$ and radius $8$ .
Explain Sam's two mistakes. Work out the correct centre and radius.
[3 marks]
Answercentre $(\dots, \dots)$ radius $\dots\dots$
D3
A circle passes through the points $(- 1, 1)$ , $(7, 1)$ and $(7, 5)$ . Work out the equation of the circle.
[4 marks]
Not drawn accurately
Answer $\dots\dots\dots\dots\dots\dots$
D4
A circle has equation $x^{2} + y^{2} + 4 x + 2 y - 20 = 0$ .
(a)
Work out the coordinates of the centre and the radius of the circle.
[3 marks]
Answercentre $(\dots, \dots)$ radius $\dots\dots$
(b)
$P$ is the point $(10, 4)$ . Show that $P$ is outside the circle, and work out the shortest distance from $P$ to the circle.
[3 marks]
Answer
Total for D4: 6 marks
D5
A circle has centre $(1, 3)$ and passes through the point $(4, - 2)$ . The point $(a, a)$ lies on the circle. Work out the two possible values of $a$ . You must show your working.
[5 marks]
Answer $a = \dots\dots$ or $a = \dots\dots$
D6
A circle has centre $(- 4, 6)$ . The $x$ -axis is a tangent to the circle.
Not drawn accurately
(a)
Write down the equation of the circle.
[2 marks]
Answer $\dots\dots\dots\dots\dots\dots$
(b)
The circle crosses the $y$ -axis at two points. Work out their exact coordinates.
[3 marks]
Answer $\dots\dots\dots$ and $\dots\dots\dots$
Total for D6: 5 marks

Extension

Stretch

No route is given. The hints are at the end of the sheet. Use one at a time.

E1
A circle touches both the $x$ -axis and the $y$ -axis. It passes through the point $(4, 2)$ . There are two possible circles. Work out the equation of each one.
[6 marks]
Hint 1 · What to try
A circle touching both axes, with $(4, 2)$ in the first quadrant, has centre $(r, r)$ and radius $r$ .
Hint 2 · The first line
Put $(4, 2)$ into $(x - r)^{2} + (y - r)^{2} = r^{2}$ .
Hint 3 · The full method
$(4 - r)^{2} + (2 - r)^{2} = r^{2}$ gives $r^{2} - 12 r + 20 = 0$ , so $r = 2$ or $r = 10$ . The circles are $(x - 2)^{2} + (y - 2)^{2} = 4$ and $(x - 10)^{2} + (y - 10)^{2} = 100$ .
Answer $\dots\dots\dots\dots$ and $\dots\dots\dots\dots$

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can write the equation of a circle from its centre and radius, and read them back.
I can complete the square to find the centre and radius of a circle.
I can find the equation of a circle through given points or from a diameter.
I can tell whether a point is inside, on or outside a circle.

Answers and mark scheme

The worked answers and the full mark scheme are free with a Mathswariz account. Teachers and students can both sign up.

Get the answers