Coordinate Geometry of the Circle
Coordinate geometry of the circle studies the equation of a circle, x^2 + y^2 = r^2 for a circle centred at the origin, or (x-a)^2 + (y-b)^2 = r^2 for a circle with centre (a, b), together with tangents, chords and points on the circumference. It builds on straight-line coordinate geometry, and questions in the AQA Level 2 paper are worth up to 9 marks, including proving a line is tangent to a circle.
Before you start
Make sure you're comfortable with these topics first:
Method
- Recall that a circle with centre (a, b) and radius r has equation (x-a)^2 + (y-b)^2 = r^2 (or x^2 + y^2 = r^2 when centred at the origin).
- To check whether a point lies on, inside or outside a circle, substitute its coordinates into the left-hand side and compare the result with r^2.
- To find a tangent at a known point on the circle, use the fact that a tangent is perpendicular to the radius at that point: find the gradient of the radius, then take the negative reciprocal.
- To decide whether a line is tangent to a circle, substitute the line's equation into the circle's equation and check that the resulting quadratic has a repeated root, so the discriminant equals 0.
- Use completing the square to rewrite an equation such as x^2 + y^2 + 2gx + 2fy + c = 0 into circle form and read off the centre and radius.
- For a circle defined by a diameter, use the midpoint formula for the centre and half the length of the diameter for the radius.
Worked example
A circle has equation x^2 + y^2 = 50. Show that the point (5, 5) lies on the circle, then find the equation of the tangent to the circle at that point.
- Substitute (5, 5): 5^2 + 5^2 = 25 + 25 = 50, which matches the equation, so (5, 5) lies on the circle.
- Find the gradient of the radius from the origin to (5, 5): gradient = 5/5 = 1.
- The tangent is perpendicular to the radius, so its gradient = -1/1 = -1.
- Substitute the point and gradient into y - 5 = -1(x - 5).
- Rearrange: y = -x + 10.
Practice questions
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Q1Write down the centre and radius of the circle x^2 + y^2 = 64.Show answer
Answer: centre (0, 0), radius 8
Q2A circle has equation (x - 3)^2 + (y + 2)^2 = 25. State its centre and radius.Show answer
Answer: centre (3, -2), radius 5
Q3Show that the point (9, 12) lies on the circle x^2 + y^2 = 225.Show answer
Answer: 9^2 + 12^2 = 81 + 144 = 225, so (9, 12) lies on the circle
Q4A circle has centre (0, 0) and passes through the point (7, 24). Find the radius.Show answer
Answer: 25 (r = sqrt(7^2 + 24^2) = sqrt(625))
Q5Determine whether the point (4, 4) lies inside or outside the circle x^2 + y^2 = 36.Show answer
Answer: inside, since 4^2 + 4^2 = 32, which is less than 36
Q6A and B are the endpoints of a diameter of a circle, where A has coordinates (-1, -2) and B has coordinates (7, 4). Find the centre and radius of the circle.Show answer
Answer: centre (3, 1), radius 5 (AB = sqrt(8^2+6^2) = 10, so radius is half of that)
Exam-style questions
Written in the style of a GCSE Further Maths exam paper, with a full mark scheme.
A circle has equation x^2 + y^2 - 6x + 4y - 12 = 0. Find the centre and radius of the circle, showing your method clearly.
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A circle has equation x^2 + y^2 = 45. Show that the line y = 2x - 15 is a tangent to the circle, and find the coordinates of the point of contact.
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A circle has equation (x + 1)^2 + (y - 3)^2 = 20. Determine whether the point (3, 6) lies inside, on, or outside the circle. Show your working.
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See real GCSE Further Maths past-paper questions, with official mark schemes →
Free printable worksheet
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