Coordinate Geometry of the Circle - Worksheets, Questions and Revision

16 original exam-style questions - 7 pages of questions with a full mark scheme - free printable PDF.

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GCSE · Geometry

G2 Coordinate Geometry of the Circle

AQA 8365 · Calculator allowed · about 100 minutes
Total Marks
Name: _______________________________    Date: ____ / ____ / ______
Answer ALL questions. Show all your working.
1
A circle has equation x2 + y2 = 81.
(a)Write down the coordinates of the centre of the circle and the radius of the circle.(2)
(b)Write down the equation of a circle with centre (0, 0) and radius 3.5(1)
(Total for Question 1 is 3 marks)
2
A circle has equation x2 + y2 = 100.
(a)Show that the point (8, 6) lies on the circle.(2)
(b)Determine whether the point (9, 5) lies inside or outside the circle. You must show working to justify your answer.(2)
(Total for Question 2 is 4 marks)
3
A circle has equation (x + 4)2 + (y - 1)2 = 36.
(a)Write down the coordinates of the centre and the radius of the circle.(2)
(b)Determine whether the point (2, 1) lies on the circle. Show your working.(2)
(Total for Question 3 is 4 marks)
4
A circle has centre (-1, 3) and passes through the point (2, -1).
(a)Calculate the radius of the circle.(3)
(b)Write down the equation of the circle in the form (x - a)2 + (y - b)2 = r2.(1)
(Total for Question 4 is 4 marks)
5
A circle has equation x2 + y2 + 8x - 2y + 8 = 0.
(a)Show that this equation can be written as (x + 4)2 + (y - 1)2 = 9.(3)
(b)Hence state the centre and radius of the circle.(2)
(Total for Question 5 is 5 marks)
6
A and B are the endpoints of a diameter of a circle, where A has coordinates (-2, -1) and B has coordinates (6, 5).
(a)Find the coordinates of the centre of the circle.(2)
(b)Find the radius of the circle.(3)
(c)Write down the equation of the circle.(1)
(Total for Question 6 is 6 marks)
7
The point P(5, 12) lies on the circle with equation x2 + y2 = 169 and centre O, the origin.
(a)Show that P lies on the circle.(1)
(b)Find the gradient of the radius OP.(1)
(c)Hence find the equation of the tangent to the circle at P, giving your answer in the form ax + by = c.(3)
(Total for Question 7 is 5 marks)
8
A circle has equation x2 + y2 - 10x + 6y + 18 = 0. Find the centre and radius of the circle, showing your method clearly.
(Total for Question 8 is 4 marks)
9
A circle has centre C(3, -2) and radius 13. A chord AB of the circle has its midpoint at M(3, 3).
(a)Find the length CM.(2)
(b)The line from the centre to the midpoint of a chord is perpendicular to the chord. Use Pythagoras' theorem to find the length of the chord AB.(3)
(Total for Question 9 is 5 marks)
10
A circle has equation x2 + y2 = 25. The line with equation y = x - 1 intersects the circle at two points.
(a)Find the coordinates of the two points of intersection.(5)
(b)Calculate the distance between the two points of intersection, giving your answer as a surd in its simplest form.(3)
(Total for Question 10 is 8 marks)
11
A circle has equation x2 + y2 = 10. A line has equation y = 3x - 10.
(a)Show that the line is a tangent to the circle.(4)
(b)Find the coordinates of the point of contact.(2)
(Total for Question 11 is 6 marks)
12
Two mobile phone masts provide circular coverage areas. Mast A is at coordinates (0, 0), measured in kilometres, and has a coverage radius of 9 km. Mast B is at coordinates (12, 5), also in kilometres, and has a coverage radius of 4 km.
(a)Calculate the distance between the two masts.(2)
(b)By comparing the distance between the masts with the sum of the two coverage radii, determine whether the coverage areas overlap, touch at exactly one point, or do not meet at all. Justify your answer.(3)
(Total for Question 12 is 5 marks)
13
A circle has equation (x - 5)2 + (y + 2)2 = 49. Exactly one of the following points lies outside the circle. Determine which one, showing your working.
(a)Which point lies outside the circle?(2)
  • A) (10, 2)
  • B) (1, -6)
  • C) (5, 4)
  • D) (-3, -2)
(Total for Question 13 is 2 marks)
14
Points A(1, 7) and B(9, 1) lie on a circle whose centre lies on the x-axis.
(a)Find the midpoint of AB.(1)
(b)Find the gradient of AB, and hence the gradient of the perpendicular bisector of AB.(2)
(c)Find the equation of the perpendicular bisector of AB.(3)
(d)The centre of the circle lies on the x-axis. Use this fact and your answer to part (c) to find the coordinates of the centre.(3)
(e)Hence find the radius of the circle, giving your answer as a surd in its simplest form.(2)
(Total for Question 14 is 11 marks)
15
A circle has equation x2 + y2 = 20. The line with equation y = 2x + b, where b > 0, is a tangent to the circle.
(a)Show that substituting the equation of the line into the equation of the circle gives 5x2 + 4bx + (b2 - 20) = 0.(2)
(b)Given that b > 0, use the condition for a tangent to find the value of b.(4)
(c)Find the coordinates of the point of contact for this value of b.(3)
(Total for Question 15 is 9 marks)
16
A, B and C are points on a circle with centre O, the origin, where A(-6, 0), B(6, 0) and C(3, 3sqrt(3)). AB is a diameter of the circle.
(a)Show that C lies on the circle x2 + y2 = 36.(2)
(b)Find the gradient of AC and the gradient of BC, leaving your answers in surd form where necessary.(3)
(c)Hence prove that angle ACB = 90 degrees.(2)
(Total for Question 16 is 7 marks)
Mark scheme · G2 Coordinate Geometry of the Circle

Question 1

Question 2

Question 3

Question 4

Question 5

Question 6

Question 7

Question 8

Question 9

Question 10

Question 11

Question 12

Question 13

Question 14

Question 15

Question 16