Circles (CIE A Level Maths: Pure 1)

Exam Questions

3 hours32 questions
1
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3 marks

Write down the equations of the circles with the following centres and radii

(i)
Centre:  (0 , 0)        Radius:  r = 4,
(ii)
Centre:  (3 , -4)       Radius:  r = 2,
(iii)

Centre:  (-5 , 0)       Radius:  r = 5.

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2
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3 marks

Write down the centre and 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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3
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4 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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4a
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2 marks
(i)
Complete the square of x squared plus 4 x.
(ii)
Complete the square of y squared minus 6 y.
4b
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4 marks
(i)
Use your answers to part (a) to show that the 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, write down the centre and the radius of the circle with equation x squared plus y squared plus 4 x minus 6 y plus 4 equals 0.

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5
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4 marks

The line segment connecting the two points (1 , 0) and (9 , 4) is the diameter of a circle.

Find the centre and radius of the circle.

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6
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3 marks

Determine if the circles with equations

open parentheses x plus 4 close parentheses squared plus y squared equals 9 space space and space open parentheses space x minus 2 close parentheses squared plus y squared equals 9

intersect once, twice or not at all.  Fully explain your answer.

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7
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2 marks

On a sketch show how a circle and a line can either have 0, 1 or 2 intersections.

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8
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5 marks

The 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 the two points of intersection.

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1
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4 marks

A circle has centre (6, -5) and goes through the point (1, 7).  Find the equation of the circle.

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2a
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2 marks

Show that x squared plus y squared plus 2 x minus 6 y plus 9 equals 0 can be written 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.

2b
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2 marks

Hence write down the centre and radius of the circle with equation x squared plus y squared plus 2 x minus 6 y plus 9 equals 0.

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3
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4 marks

The linespace x plus y equals negative 7 spacemeets 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 the point of intersection.

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4
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4 marks

The linespace 7 x plus y equals negative 6 spaceintersects the circle space open parentheses x minus 2 close parentheses squared plus open parentheses y minus 5 close parentheses squared equals 25 spaceat the points Aand B. Find the coordinates of Aand B.

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5a
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4 marks

A circle C has centre (-4, 1) and passes through the point P(0, 3)

Find an equation for the circle C.

5b
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3 marks

Find an equation for the tangent to the circle at P.

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6a
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2 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.

6b
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1 mark

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

6c
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4 marks

Hence find the equation of the circle.

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7
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6 marks

C subscript 2Circles 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, and 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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8a
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1 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 be in the form y equals m x, where m is the gradient of the line.

8b
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4 marks

Show that the gradients of l subscript 1 and l subscript 2 must be the solutions to the equation

19 m squared plus 84 m plus 32 equals 0.

8c
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2 marks

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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1
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5 marks

The points A (-3, 1) and B (3, -7) are the two endpoints of the diameter AB of a circle. Find the equation of the circle.

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2a
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2 marks

Show that x squared plus y squared plus 5 x minus 2 y minus 5 equals 0 can be written 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.

2b
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2 marks

Hence write down the centre and radius of the circle with equation x squared plus y squared plus 5 x minus 2 y minus 5 equals 0.

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3
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4 marks

The line space y plus 2 x equals 11 spacemeets 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 the point of intersection.

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4
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4 marks

The line x plus 5 y plus 22 equals 0 intersects the circle space x squared plus y squared plus 4 x plus 8 y minus 6 equals 0 spaceat the points A and B.  Find the coordinates of Aand B

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5a
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4 marks

A circle C has centre (-2, 3) and passes through the point P(6, -3)

Find an equation for the circle C.

5b
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3 marks

Find an equation for the tangent to the circle at P.

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6a
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2 marks

The points A left parenthesis negative 3 comma space 6 right parenthesis comma space B left parenthesis 5 comma space minus 4 right parenthesis and C left parenthesis 6 comma space 5 right parenthesis lie on a circle.

Show that angle A C B equals 90 degree. 

6b
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1 mark

Deduce a geometrical property of the line segment AB.

6c
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4 marks

Hence find the equation of the circle.

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7a
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2 marks

Triangle A B C has vertices A(-8, 1), B(12, 16) and C(12, 1).  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

Write down the coordinates of points R and Q.

7b
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5 marks

Find the coordinates of point P.

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8
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7 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 (0, 14).

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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1
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6 marks

The points A(2, -21) and B(-5, 3) are the two endpoints of the diameter A Bof a circle. Find the equation of the circle in the form a x squared plus a y squared plus b x plus c y plus d equals 0, where a comma b comma c and d are integers to be found.

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2
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4 marks

Find the centre and radius of the circle with equation x squared plus y squared plus x minus 3 y plus 2 equals 0.

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3
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7 marks

The 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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4
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4 marks

The points A(-2, 3), B(0, 6) and C(k, -1) lie on a circle, where B C is the diameter of the circle.

Find the value of k.

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5
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7 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 the circle, and the tangent to the circle at point P has a gradient of -3. Find the two possible sets of coordinates for point P.

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6
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7 marks

The pointsspace A left parenthesis 4 comma space 6 right parenthesis comma space B left parenthesis 7 comma space 2 right parenthesis spaceand C left parenthesis 12 comma space 12 right parenthesis lie on a circle.

Find the equation of the circle.

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7
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8 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 (5, 0).

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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8
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11 marks

The diagram below shows circles C subscript 1 and C subscript 2 which intersect at the two 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, and points A and B lie along the line with equation  3 x minus y equals negative 1. Circle space C subscript 2 spacealso passes through the point (-13, 2)

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

Find an equation of circle C subscript 2.

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