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Circles (Edexcel A Level Maths: Pure): Exam Questions

Exam code: 9MA0

4 hours37 questions
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Very Hard
13 marks

Find the equations of the following circles

(i) Centre: open parentheses 0 comma space 0 close parentheses, radius of 4

(ii) Centre: open parentheses 3 comma space minus 4 close parentheses, radius of 2

(iii) Centre: open parentheses negative 5 comma space 0 close parentheses, radius of 5

How did you do?

23 marks

Find the coordinates of the centre and the length of the radius for each of the following circles

(i) x squared plus y squared equals 5 squared

(ii) open parentheses x plus 3 close parentheses squared plus open parentheses y minus 2 close parentheses squared equals 49

(iii) x squared plus open parentheses y plus 4 close parentheses squared equals 144

How did you do?

33 marks

On separate diagrams, sketch the circles with the following equations

(i) x squared plus y squared equals 9

(ii) open parentheses x minus 4 close parentheses squared plus open parentheses y minus 3 close parentheses squared equals 4 squared

How did you do?

4a2 marks

Complete the square of the following expressions

(i) x squared plus 4 x

(ii) y squared minus 6 y

How did you do?

4b3 marks

(i) Use your answers to part (a) to show that the circle with equation

  x squared plus y squared plus 4 x minus 6 y plus 4 equals 0 

can be written in the form 

open parentheses x plus 2 close parentheses squared plus open parentheses y minus 3 close parentheses squared equals 9

(ii) Hence, find the coordinates of the centre and the length of the radius of the circle.

How did you do?

53 marks

The line segment between the two points open parentheses 1 comma space 0 close parenthesesand open parentheses 9 comma space 4 close parentheses is the diameter of a circle.

Find the coordinates of the centre of the circle and the exact radius.

How did you do?

64 marks

The straight line with equation y equals x minus 1 intersects the circle with equation

open parentheses x minus 5 close parentheses squared plus open parentheses y minus 4 close parentheses squared equals 18

at two distinct points.

Find the coordinates of these two points.

How did you do?

74 marks

The straight linespace 7 x plus y equals negative 6 spaceintersects the circle with equation

space open parentheses x minus 2 close parentheses squared plus open parentheses y minus 5 close parentheses squared equals 25 space

at the points Aand B.

Find the coordinates of Aand B.

How did you do?

82 marks

Determine if the circles with equations

open parentheses x plus 4 close parentheses squared plus y squared equals 9

open parentheses space x minus 2 close parentheses squared plus y squared equals 9

intersect once, twice or not at all, giving a reason for your answer.

How did you do?

9a2 marks

Express

x squared plus y squared plus 2 x minus 6 y plus 9 equals 0

in the form

open parentheses x minus a close parentheses squared plus open parentheses y minus b close parentheses squared equals r squared

where a, b and r are integers to be found.

How did you do?

9b2 marks

Hence find the radius and the coordinates of the centre of the circle with equation

x squared plus y squared plus 2 x minus 6 y plus 9 equals 0

How did you do?

1a3 marks

A circle has equation

x squared plus y squared minus 10 x plus 16 y equals 80

Find

(i) the coordinates of the centre of the circle,

(ii) the radius of the circle.

How did you do?

1b
Sme Calculator2 marks

Given that P is the point on the circle that is furthest away from the origin O, find the exact length O P

How did you do?

23 marks

A circle has centre open parentheses 6 comma space minus 5 close parentheses and passes through the point open parentheses 1 comma space 7 close parentheses. 

Find the equation of the circle.

How did you do?

34 marks

The line x plus y equals negative 7 meets the circle with equation

left parenthesis x minus 1 right parenthesis squared plus left parenthesis y minus 2 right parenthesis squared equals 50

(i) Show that the line and circle meet at one point only.

(ii) Find the coordinates of this point.

How did you do?

4a3 marks

A circle C has a centre with coordinates open parentheses negative 4 comma space 1 close parentheses and passes through the point P with coordinates open parentheses 0 comma space 3 close parentheses.

Find an equation for C.

How did you do?

4b4 marks

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

Give your answer in the form y equals a plus b x.

How did you do?

5a3 marks

The points A left parenthesis 3 comma space 5 right parenthesis comma space B left parenthesis 5 comma space 3 right parenthesis and C left parenthesis 9 comma space 7 right parenthesis lie on a circle.

Show that triangle A B C is a right-angle triangle.

How did you do?

5b1 mark

Explain why the line segment A C must be the diameter of the circle.

How did you do?

5c4 marks

Find the equation of the circle.

How did you do?

65 marks

Circles C subscript 1, C subscript 2 and C subscript 3 all have their centres on the x-axis.  

  • Circle C subscript 1 has equation open parentheses x plus 7 close parentheses squared plus y squared equals 4

  • Circle C subscript 3 has equation x squared plus y squared minus 10 x plus 16 equals 0

  • Circles C subscript 1 and C subscript 2 touch at point A

  • Circles C subscript 2 and C subscript 3 touch at point B.

q7-3-2-circles-medium-a-level-maths-pure-screenshot

Find the coordinates of the centre of circle C subscript 2.

How did you do?

7a1 mark

A circle has equation x squared plus y squared minus 12 x plus 14 y equals negative 68.

The lines l subscript 1 and l subscript 2 are both tangents to the circle, and they intersect at the origin.

q8-3-2-circles-medium-a-level-maths-pure-screenshot

Explain why the equations for l subscript 1 and l subscript 2 must each have the form y equals m x.

How did you do?

7b4 marks

Show that m satisfies the equation

19 m squared plus 84 m plus 32 equals 0

How did you do?

7c1 mark

Hence find the equations of l subscript 1 and l subscript 2, giving your answers in the form space y equals m x.

How did you do?

84 marks

The line y plus 2 x equals 11meets the circle with equation

x squared plus y squared plus 6 x minus 14 y equals negative 38

(i) Show that the line and circle meet at one point only.

(ii) Find the coordinates of this point.

How did you do?

9a3 marks

Express

x squared plus y squared plus 5 x minus 2 y minus 5 equals 0

in the form

open parentheses x minus a close parentheses squared plus open parentheses y minus b close parentheses squared equals r squared

where a comma space b and r are constants to be found.

How did you do?

9b2 marks

Hence find the radius and the coordinates of the centre of the circle with equation

x squared plus y squared plus 5 x minus 2 y minus 5 equals 0

How did you do?

10
Sme Calculator4 marks

The points A open parentheses negative 3 comma space 1 close parentheses and B open parentheses 3 comma space minus 7 close parentheses are the endpoints of the diameter A B of a circle.

Find an equation for the circle.

How did you do?

115 marks

The straight line x plus 5 y plus 22 equals 0 intersects the circle with equation

space x squared plus y squared plus 4 x plus 8 y minus 6 equals 0 space

at the points A and B. 

Find the coordinates of Aand B

How did you do?

12a3 marks

A circle C has a centre with coordinates open parentheses negative 2 comma space 3 close parentheses and passes through the point P with coordinates open parentheses 6 comma space minus 3 close parentheses.

Find an equation for C.

How did you do?

12b4 marks

Find the equation of the tangent to C at P.

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

How did you do?

134 marks

Find the exact radius and the coordinates of the centre of the circle with equation

x squared plus y squared plus x minus 3 y plus 2 equals 0

How did you do?

1a2 marks

The circle C has equation

x squared plus y squared minus 6 x plus 10 y plus k equals 0

where k is a constant.

Find the coordinates of the centre of C.

How did you do?

1b3 marks

Given that C does not cut or touch the x-axis, find the range of possible values for k.

How did you do?

2a3 marks
A line intersects the x axis at a negative value and the y axis at a positive value. The line is tangent to circle C with centre in the first quadrant. The circle does not intersect any axes.
Figure 3

The circle C has centre A with coordinates open parentheses 7 comma space 5 close parentheses.

The line l, with equation y equals 2 x plus 1, is the tangent to C at the point P, as shown in Figure 3.

Show that an equation of the line P A is 2 y plus x equals 17.

How did you do?

2b4 marks

Find an equation for C.

How did you do?

2c3 marks

The line with equation y equals 2 x plus k comma space space space space k not equal to 1 is also a tangent to C.

Find the value of the constant k.

How did you do?

3a4 marks

The circle C has equation

x squared plus y squared minus 10 x plus 4 y plus 11 equals 0

Find

(i) the coordinates of the centre C,

(ii) the exact radius of C, giving your answer as a simplified surd.

How did you do?

3b
Sme Calculator5 marks

The line L has equation y equals 3 x plus k where k is a constant.

Given that L is a tangent to C, find the possible values of k, giving your answers as simplified surds.

How did you do?

4a2 marks
Graph with axes marked x and y. A line labelled l through the origin is tangent to a circle at point P. The circle's centre is marked as N, and the circle is entirely in the first quadrant.
Figure 4

Figure 4 shows a sketch of a circle C with centre N left parenthesis 7 comma space 4 right parenthesis

The line l with equation y equals 1 third x is a tangent to C at the point P.

Find the equation of line P N in the form y equals m x plus c, where m and c are constants,

How did you do?

4b4 marks

Find an equation for C.

How did you do?

4c
Sme Calculator3 marks

The line with equation y equals 1 third x plus k, where k is a non-zero constant, is also a tangent to C.

Find the value of k.

How did you do?

56 marks

The straight line x plus y equals c intersects the circle

x squared plus y squared minus 6 x plus 10 y minus 16 equals 0

at exactly two points. 

Find the range of possible values of c.

How did you do?

66 marks

The points A open parentheses 2 comma space minus 21 close parentheses and B open parentheses negative 5 comma space 3 close parentheses are the two endpoints of the diameter A Bof a circle.

Find the equation of the circle in the form

x squared plus y squared plus a x plus b y plus c equals 0

where a comma space b and c are integers to be found.

How did you do?

7a4 marks

Triangle A B C has vertices A open parentheses negative 8 comma space 1 close parentheses, B open parentheses 12 comma space 16 close parentheses and C open parentheses 12 comma space 1 close parentheses.

A circle with equation open parentheses x minus 7 close parentheses squared plus open parentheses y minus 6 close parentheses squared equals 25 touches triangle A B C at the three points P comma space Qand R, as shown in the diagram below.

q7-3-2-circles-hard-a-level-maths-pure-screenshot

Find the coordinates of points R and Q.

How did you do?

7b6 marks

Find the coordinates of point P.

How did you do?

84 marks

The points A open parentheses negative 2 comma space 3 close parentheses, B open parentheses 0 comma space 6 close parentheses and C open parentheses k comma space minus 1 close parentheses lie on a circle, where B C is the diameter of the circle.

Find the value of k.

How did you do?

97 marks

A circle C has equation

x squared plus y squared minus 10 x minus 4 y plus 19 equals 0

Point P lies on C, and the tangent to C at P has a gradient of negative 3.

Find the two possible sets of coordinates for point P.

How did you do?

1a3 marks

A circle C spacehas equation

x squared plus y squared plus 6 k x minus 2 k y plus 7 equals 0

where k is a constant.

Find in terms of k,

(i) the coordinates of the centre of C

(ii) the radius of C

How did you do?

1b
Sme Calculator6 marks

The line with equationspace y equals 2 x – 1 intersects C at 2 distinct points.

Find the range of possible values of k.

How did you do?

2a3 marks
Graph using x and y axes. The graph shows a circle labelled C and a straight line l below with a negative gradient. The straight line crosses the negative x and y axes.
Figure 3

Figure 3 shows the circle C with equation

x squared plus y squared minus 10 x minus 8 y plus 32 equals 0

and the line l with equation

2 y plus x plus 6 equals 0

Find

(i) the coordinates of the centre of C,

(ii) the radius of C

How did you do?

2b
Sme Calculator5 marks

Find the shortest distance between C and l.

How did you do?

3a3 marks

A circle C with radius r

  • lies only in the 1st quadrant

  • touches the x-axis and touches the y-axis

The line l has equation 2 x plus y equals 12

Show that the x coordinates of the points of intersection of l with C satisfy

5 x squared plus open parentheses 2 r minus 48 close parentheses x plus open parentheses r squared minus 24 r plus 144 close parentheses equals 0

How did you do?

3b
Sme Calculator4 marks

Given also that l is a tangent to C, find the two possible values of r, giving your answers as fully simplified surds.

How did you do?

47 marks

A circle has equation

x squared plus y squared plus 14 x minus 6 y equals negative 41

The lines l subscript 1and l subscript 2 are both tangents to the circle, and they intersect at the point open parentheses 0 comma space 14 close parentheses.

q8-3-2-circles-hard-a-level-maths-pure-screenshot

Find the equations of l subscript 1 and l subscript 2, giving your answers in the form y equals m x plus c.

How did you do?

57 marks

A circle has equation

x squared plus y squared plus 4 x plus 12 y equals negative 23

The lines l subscript 1 and l subscript 2 are both tangents to the circle, and they intersect at the point open parentheses 5 comma space 0 close parentheses.

q7-3-2-circles-vhard-a-level-maths-pure-screenshot

Find the equations of l subscript 1 and l subscript 2, giving your answers in the form y equals m x plus c.

How did you do?

611 marks

The diagram below shows circles C subscript 1 and C subscript 2 intersecting at the points A and B.

  • Circle C subscript 1 has equation  x squared plus y squared minus 16 x minus 10 y plus 39 equals 0

  • Points A and B lie along the line with equation  3 x minus y equals negative 1

  • Circle space C subscript 2 spacepasses through the point open parentheses negative 13 comma space 2 close parentheses

HFlQzKQw_q8-3-2-circles-medium-a-level-maths-pure-screenshot

Find an equation of circle C subscript 2.

How did you do?