Question 1 Easy
State the centre of the circle (x - 4)^2 + (y - 1)^2 = 16.
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Answer: (4, 1)
- Set each bracket to zero.
- x = 4 and y = 1.
- Method (1): Set each bracket to zero.
- Accuracy (1): Answer (4, 1)
Equation of a circle practice questions with answers
A second set of equation of a circle questions for GCSE Maths. They are different from the ones on the topic page, so work through both. Try each question, then open the answer to check your working.
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Question 2 Easy
State the centre of the circle (x + 3)^2 + (y - 7)^2 = 25.
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Answer: (-3, 7)
- x + 3 = 0 gives x = -3.
- y - 7 = 0 gives y = 7.
- Method (1): x + 3 = 0 gives x = -3.
- Accuracy (1): Answer (-3, 7)
Question 3 Medium
The circle with centre (2, 1) passes through (5, 5). Work out r squared.
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Answer: 25
- The steps are 3 and 4.
- r squared is 9 + 16.
- Method (1): The steps are 3 and 4.
- Accuracy (1): Answer 25
Question 4 Medium
A circle has its centre at the origin and passes through (-5, 12). Write its equation.
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Answer: x^2+y^2=169
- r squared is 25 + 144.
- The centre is (0, 0), so there are no brackets.
- Method (1): r squared is 25 + 144.
- Accuracy (1): Answer x^2+y^2=169
Question 5 Hard
The ends of a diameter are (-1, 2) and (5, 10). Write the equation of the circle.
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Answer: (x-2)^2+(y-6)^2=25
- The centre is the midpoint (2, 6).
- The steps to an end are 3 and 4.
- Method (1): The centre is the midpoint (2, 6).
- Accuracy (1): Answer (x-2)^2+(y-6)^2=25
Question 6 Hard
A circle has centre (3, 3) and passes through the origin. Write its equation.
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Answer: (x-3)^2+(y-3)^2=18
- The steps from the centre to the origin are 3 and 3.
- r squared is 9 + 9.
- Method (1): The steps from the centre to the origin are 3 and 3.
- Accuracy (1): Answer (x-3)^2+(y-3)^2=18
Exam-style question
A longer question with marks, in the style of a written paper. This is a Yumio practice question, not a past paper question.
Exam-style question 13 marks
Show whether the point (5, 5) is on, inside or outside the circle x^2 + y^2 = 36.
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Answer: outside
- Substitute x = 5 and y = 5.
- 25 + 25 = 50.
- 50 is bigger than 36, so the point is outside.
- Method (1): Substitutes the coordinates into x^2 + y^2
- Accuracy (1): Gets 50
- Accuracy (1): Compares with 36 and says outside
Quick checks
Short one-step questions. Pick an option before you open the answer.
Check 1
What is the equation of a circle with centre (0, 0) and radius r?
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x^2 + y^2 = r^2. The right-hand side is the radius squared. x^2 + y^2 = r^2. The right-hand side is the radius squared.
Check 2
What is the equation of a circle with centre (a, b) and radius r?
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(x - a)^2 + (y - b)^2 = r^2. (x - a)^2 + (y - b)^2 = r^2.
Check 3
How do you read the centre from (x + 4)^2 + (y - 1)^2 = 9?
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Set each bracket to zero: x = -4 and y = 1, so the centre is (-4, 1). Set each bracket to zero: x = -4 and y = 1, so the centre is (-4, 1).
Check 4
The equation ends in 49. What is the radius?
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7. Take the square root of the radius squared. 7. Take the square root of the radius squared.
Check 5
How do you find r squared from a centre and a point on the circle?
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Square the horizontal step, square the vertical step and add. Square the horizontal step, square the vertical step and add.
Check 6
Why is the right-hand side r squared and not r?
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The equation comes from Pythagoras, where the distance squared is the sum of two squares. The equation comes from Pythagoras, where the distance squared is the sum of two squares.
Check 7
What is the centre of the circle with a diameter from (2, 1) to (8, 9)?
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(5, 5). It is the midpoint of the diameter. (5, 5). It is the midpoint of the diameter.
Check 8
How can you check an equation you wrote from a diameter?
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Substitute the other end of the diameter. It must give the right-hand side. Substitute the other end of the diameter. It must give the right-hand side.
Check 9
A point gives a value smaller than r squared. Where is it?
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Inside the circle. Inside the circle.
Check 10
A point gives a value larger than r squared. Where is it?
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Outside the circle. Outside the circle.
Check 11
What does x^2 + (y - 3)^2 = 16 tell you about the centre?
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The x-coordinate of the centre is 0, so the centre is (0, 3) and the radius is 4. The x-coordinate of the centre is 0, so the centre is (0, 3) and the radius is 4.
Check 12
Why do the signs in the brackets look reversed?
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Each bracket subtracts the coordinate of the centre, so a negative coordinate becomes a plus. Each bracket subtracts the coordinate of the centre, so a negative coordinate becomes a plus.
Check 13
What is the equation of the circle with centre (0, 0) through the point (3, 4)?
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x^2 + y^2 = 25, because 9 + 16 = 25. x^2 + y^2 = 25, because 9 + 16 = 25.
Check 14
How do you find where a circle meets the x-axis?
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Put y = 0 in the equation and solve for x, keeping both square roots. Put y = 0 in the equation and solve for x, keeping both square roots.
Check 15
Do you need to square-root before writing the equation?
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No. You need r squared, so leave the sum of squares as it is. No. You need r squared, so leave the sum of squares as it is.
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