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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

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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

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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

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4a2 marks

Complete the square of the following expressions

(i) x squared plus 4 x

(ii) y squared minus 6 y

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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.

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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.

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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.

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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.

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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.

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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.

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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

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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.

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1b
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Given that P is the point on the circle that is furthest away from the origin O, find the exact length O P

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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.

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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.

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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.

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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.

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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.

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5b1 mark

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

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5c4 marks

Find the equation of the circle.

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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.

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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.

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7b4 marks

Show that m satisfies the equation

19 m squared plus 84 m plus 32 equals 0

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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.

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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.

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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.

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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

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10
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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.

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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

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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.

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12b4 marks

Find the equation of the tangent to C at P.

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

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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

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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.

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1b3 marks

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

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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.

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2b4 marks

Find an equation for C.

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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.

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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.

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3b
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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.

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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,

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4b4 marks

Find an equation for C.

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4c
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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.

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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.

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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.

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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.

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7b6 marks

Find the coordinates of point P.

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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.

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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.

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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

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

Find the range of possible values of k.

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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

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2b
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Find the shortest distance between C and l.

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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

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3b
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Given also that l is a tangent to C, find the two possible values of r, giving your answers as fully simplified surds.

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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.

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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.

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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.

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