Equation of a Circle (AQA GCSE Maths)

Exam Questions

2 hours28 questions
1
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1 mark

The equation of a circle is    x squared space plus space y squared space equals space 11

Work out the length of the diameter.
Circle your answer.

square root of 11 2 square root of 11 square root of 22 22

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2
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1 mark

The equation of a circle isspace x squared space plus space y squared space equals space 9 space

Work out the length of the diameter.
Circle your answer.

3 6 9 18

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3
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2 marks

Describe fully the graph which has the equation x2 + y2 = 9.

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4
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1 mark

A circle, centre O, passes through (5, 0).

q22-paper-2h-june-2018-aqa-gcse-maths

What is the equation of the circle?

Circle your answer.

x squared space plus space y squared space equals space 25 x squared space plus space y squared space equals space 5 x squared space plus space y squared space equals space 10 x squared space plus space y squared space equals space 100

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5
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1 mark

Work out the diameter of the circle x squared space plus space y squared space equals space 64

Circle your answer.

8 16 32 128

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1a
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2 marks

Construct the graph of space x squared space plus space y squared equals space 9

q2a-medium-2-18-equation-of-circle-edexcel-gcse-maths

1b
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3 marks

By drawing the line x space plus space y space equals space 1 on the grid, solve the equations x squared space plus space y squared space equals space 9
x space plus space y space equals 1

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2a
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1 mark

The equation of a curve is y space equals space a to the power of x
A is the point where the curve intersects the y-axis.

State the coordinates of A.

2b
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3 marks

The equation of circle bold C is x squared space plus space y squared space equals space 16

The circle bold C is translated by the vector open parentheses table row 3 row 0 end table close parentheses to give circle bold B.

Draw a sketch of circle bold B.

Label with coordinates
   the centre of circle bold B
   and any points of intersection with the x-axis.

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3a
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2 marks

On the grid, construct the graph of space x squared space plus space y squared space equals space 16

q3a-medium-2-18-equation-of-circle-edexcel-gcse-maths

3b
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3 marks

Find estimates for the solutions of the simultaneous equations

table attributes columnalign right center left columnspacing 0px end attributes row cell x squared space plus space y squared space end cell equals cell space 16 end cell row cell y space end cell equals cell space 2 x space plus 1 end cell end table

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4a
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1 mark

A comma space B and C are points on the circle    x squared space plus space y squared space equals space 36   as shown.

A is on the y-axis.
B is on the x-axis.
M is the midpoint of A B.
C O M is a straight line.

q28-paper1h-nov2017-aqa-gcse-maths

Show that the coordinates of A are (0, 6).

4b
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1 mark

Work out the coordinates of B.

4c
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2 marks

Show that the equation of the straight line passing through space C comma space O spaceand M is y space equals space x.

4d
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3 marks

Work out the coordinates of C.
Give your answers in surd form.

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5
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4 marks

P(–1, 4) is a point on a circle, centre O

q27-paper1h-june2017-aqa-gcse-maths

Work out the equation of the tangent to the circle at P.

Give your answer in the form    y space equals space m x space plus space c.

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6a
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2 marks

Here is a sketch of the circle  begin mathsize 16px style x squared plus y squared equals 45 end style

q18-paper4-nov2020-ocr-gcse-maths

Show that the tangent to this circle at the point (-3, 6) has a gradient of 1 half

6b
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2 marks

Find the equation of the tangent at the point (-3, 6).

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7a
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4 marks

The point ( -5, 2 ) lies on the circumference of a circle, centre (0, 0).

Find the equation of the circle.

7b
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2 marks

Work out the gradient of the tangent to the circle at (-5, 2).

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8a
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2 marks

In this question all units are in cm.

A circle has equation x squared plus y squared equals 36

Write down the radius and centre of the circle.

radius: .......................................................... cm
centre: (..........................., ...........................)     

8b
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4 marks

The distinct points A (a comma space square root of 11) and B (b comma space square root of 11) lie on the circumference of the circle.

Work out the length AB.

................................................... cm 

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1
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3 marks

Here is a circle, centre O, and the tangent to the circle at the point space P(4, 3) on the circle.

q1-hard-2-18-equation-of-circle-edexcel-gcse-maths

Find an equation of the tangent at the point P.

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2a
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2 marks

On the grid, draw the graph of x squared space plus space y squared space equals 12.25

q2a-hard-2-18-equation-of-circle-edexcel-gcse-maths

2b
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3 marks

Hence find estimates for the solutions of the simultaneous equation

table row cell x squared space plus y squared space end cell equals cell space 12.25 space end cell row cell 2 x plus y end cell equals 1 end table

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3
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3 marks

bold L is the circle with equation x squared space plus space y squared space equals 4

P open parentheses 3 over 2 comma space fraction numerator square root of 7 over denominator 2 end fraction close parentheses is a point on bold L.

Find an equation of the tangent to bold L at the point P.

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4a
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1 mark

The diagram shows a circle, centre the origin.

q4-hard-2-18-equation-of-circle-edexcel-gcse-maths

Write down the equation of the circle.

4b
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2 marks

Point straight P has coordinates (8, —6).
Show that point straight P lies on the circle.

4c
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5 marks

Find the equation of the tangent to the circle at point straight P.

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5a
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3 marks

A circle has equation x2 + y2 = 80. 

Calculate the diameter of the circle.
Give your answer as a surd in its simplest form.

5b
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2 marks

Colin says that the point (5, 7) lies outside the circle.

Is Colin correct?
Show your reasoning.

5c
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6 marks

Show that the line with equation y = space 1 half x + 10 is a tangent to the circle.

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6
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5 marks

P is a point on the circle with equation space x squared space plus space y squared space equals space 80

P has x-coordinate 4 and is below the x-axis.

q28-paper-3h-june-2018-aqa-gcse-maths

Work out the equation of the tangent to the circle at P.

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7
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5 marks

The diagram shows the circle    x squared space plus space y to the power of 2 space end exponent equals space 10

P lies on the circle and has x-coordinate 1
The tangent atspace P space intersects the x-axis at Q.

q28-paper1h-specimen2015-aqa-gcse-maths

Work out the coordinates of Q.

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8
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5 marks

The diagram shows a circle with centre (0, 0) and a tangent at the point (–12, 16).

q14-paper4-nov2019-ocr-gcse-maths

The tangent crosses the y-axis at the point (0, p).

Find the value of p.

p = .....................................................

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1a
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4 marks

The diagram shows a circle, centre O.

q1-vhard-2-18-equation-of-circle-edexcel-gcse-maths

The circumference of the circle is 20π cm.
Find the equation of the circle.

1b
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4 marks

The line 10x + py = q is a tangent at the point (5, 4) in another circle with centre (0, 0).

Find the value of p and the value of q.

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2
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5 marks

The line l is a tangent to the circle x squared space plus space y squared space equals space 40 at the point A.
A is the point (2, 6).

The line l crosses the x-axis at the point P.

Work out the area of triangle O A P.

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3a
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2 marks

The diagram shows the circle x2 + y2 =5.

q3-vhard-2-18-equation-of-circle-edexcel-gcse-maths

Mandy says that the point (2, 1.5) lies inside the circle.

Is she correct?
Show how you decide.

3b
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6 marks

The tangent to the circle at the point P (-2, 1) intersects the y-axis at A.

Show that the area of the triangle APO is 5 square units.

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4
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4 marks

The diagram shows a circle, centre O.

q4-vhard-2-18-equation-of-circle-edexcel-gcse-maths

A B is the tangent to the circle at the point A.
Angle O B A space equals space 30 to the power of ring operator

Point B has coordinates (16, 0)
Point P has coordinates (3p, p)

Find the value of p.
Give your answer correct to 1 decimal place.
You must show all your working.

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5
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5 marks

Prove algebraically that the straight line with equation x space minus space 2 y space equals 10 is a tangent to the circle with equation x squared space plus space y squared space equals space 20

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6
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5 marks

In this question, all lengths are in centimetres.

A is a point on a circle, centre O.
B is a point on a different circle, centre O.
A B space equals space 20

q27-paper-3h-june-2019-aqa-gcse-maths

The equation of the larger circle is    x to the power of 2 space end exponent plus space y to the power of 2 space end exponent equals space 144

radius of smaller circle : radius of larger circle   = 4 : 5

Work out the size of angle A O B

............................................ degrees

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7a
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2 marks

P (4, 8) is a point on a circle, centre O.

The tangent at P intersects the axes at points A and B.

q25-paper-1h-june-2019-aqa-gcse-maths

Show that the gradient of the tangent is negative 1 half

7b
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4 marks

Work out the length A B.

Give your answer in the form space a square root of space 5 space end root where a is an integer.

You must show your working.

....................................units

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