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Grade 8 · Higher · free GCSE maths worksheet

Equation of a circle centred at the origin

x^2 + y^2 = r^2.

13 original questions, 39 marks. Print the worksheet, mark it with the answers sheet, and use the solutions to see every step written out by hand.

Worksheet PDFWorked solutions PDFAnswers PDF

The questions

1

A circle has its centre at the origin O and has radius 7.

Write down the equation of the circle.

1 mark
Show answer
x2 + y2 = 49
2

A circle has equation x2 + y2 = 81

Write down the radius of the circle.

1 mark
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9
3

A circle has its centre at the origin O. The diameter of the circle is 16.

Find the equation of the circle.

2 marks
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x2 + y2 = 64
4

On the grid, draw the circle with equation x2 + y2 = 25.

−6−5−4−3−2−1123456−6−5−4−3−2−1123456Oxy
2 marks
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A circle with centre (0, 0) passing through (5, 0), (-5, 0), (0, 5) and (0, -5)
5

A circle has equation x2 + y2 = 45.

For each point, state whether it lies inside the circle, on the circle or outside the circle. Show your working.

  1. P (6, 3)
  2. Q (5, −4)
  3. R (−7, 1)
3 marks
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a) on the circle b) inside c) outside
6

The circle C has equation x2 + y2 = 61

  1. Show that the point A (5, −6) lies on C.
  2. Does the point B (4, 7) lie on C? You must justify your answer.

3 marks
Show answer
a) 25 + 36 = 61, so A lies on the circle b) No: 16 + 49 = 65, which is not 61, so B does not lie on the circle
7

A circle has equation x2 + y2 = 100.

  1. A point on the circle has x-coordinate 6 and a positive y-coordinate. Work out its y-coordinate.
  2. A point on the circle has y-coordinate −8. Work out both possible x-coordinates.
3 marks
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a) 8 b) 6 and -6
8

The point (k, 2k) lies on the circle with equation x2 + y2 = 45.

Work out the two possible values of k.

3 marks
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k = 3 or k = -3
9

A circle has centre O, the origin, and passes through the point T (5, 12).

  1. Find an equation of the circle.
  2. The point D has coordinates (10, 9). Decide whether D is inside the circle, on the circle, or outside the circle. You must show your working.

4 marks
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a) x2 + y2 = 169 b) outside the circle, because 100 + 81 = 181 > 169
10

On the grid, draw the circle x2 + y2 = 25 and the straight line y = x + 1.

Use your graphs to solve the simultaneous equations

x2 + y2 = 25   and   y = x + 1

−6−5−4−3−2−1123456−6−5−4−3−2−1123456Oxy
4 marks
Show answer
x = 3, y = 4 and x = -4, y = -3
11

The point A (3, 4) lies on the circle with equation x2 + y2 = 25. The centre of the circle is O (0, 0).

Find the equation of the tangent to the circle at A. Give your answer in the form ax + by = c where a, b and c are integers.

4 marks
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3x + 4y = 25
12

A circle has equation (x − 4)2 + (y − 3)2 = 25. Its centre is (4, 3) and its radius is 5.

  1. On the grid, draw the circle.
  2. Work out the coordinates of the points where the circle crosses the x-axis.
−2−112345678910−3−2−1123456789Oxy
4 marks
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a) Circle centre (4, 3), radius 5, through (0, 0), (8, 0) and (0, 6) b) (0, 0) and (8, 0)
13

The line with equation y = x + 2 intersects the circle with equation x2 + y2 = 34 at the points A and B.

Find the coordinates of A and B.

5 marks
Show answer
(-5, -3) and (3, 5)

Every question here is original and written by Robinson Tuition. The grade follows the Edexcel grade descriptors and suits AQA and OCR too; this is not an exam-board paper.

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