About this chapter
This chapter is Pearson's coordinate geometry of the circle. Most students can find a centre and radius once the equation is in the right form; the marks that separate grades come from what follows, such as a tangent at a point, a line that only just touches, or two circles that never meet.
It covers the equation of a circle, completing the square to find the centre and radius, tangents and chords, the angle in a semicircle, and lines meeting circles through the discriminant. The later questions add a region cut off by a chord, found with radians from Pure 1, and a kite formed by two tangents drawn from an outside point.
The ten questions
- Q1 (4 marks): the circle with diameter from A(−1, 4) to B(5, −4).
- Q2 (5 marks): complete the square to find the centre and radius, then show the circle touches the x-axis.
- Q3 (6 marks): the radius and equation from a centre and a point, then the tangent at that point.
- Q4 (5 marks): where y = x + 1 meets x² + y² = 25, found algebraically, and the exact chord length.
- Q5 (5 marks): the perpendicular bisector of a chord leads to a centre on the x-axis and the circle.
- Q6 (8 marks): centres and radii of two circles, then show they do not meet.
- Q7 (8 marks): y = 2x + k as a tangent: an equation in x, the two values of k, and the point of contact.
- Q8 (8 marks): a right angle at C means AB is a diameter, then the circle and the other end of a diameter.
- Q9 (9 marks): a line cuts a circle at A and B, the angle at the centre is π/2, then the exact area of the segment.
- Q10 (10 marks): two tangents from T(14, 6): the tangent length, the kite's area, an angle in radians, and the region outside the circle.
Key skills tested
Equation of a circle. Centre (a, b) and radius r give (x−a)² + (y−b)² = r².
Centre and radius. Complete the square in x and in y. For example, x² + y² − 4x + 2y = 4 becomes (x−2)² + (y+1)² = 9, with centre (2, −1) and radius 3.
Midpoints and distances. The centre is the midpoint of any diameter, and the radius is the distance from the centre to any point on the circle.
Tangents. A tangent is perpendicular to the radius at the point of contact, so its gradient is the negative reciprocal of the radius's gradient.
Chords. The perpendicular bisector of any chord passes through the centre.
Angle in a semicircle. If angle ACB is 90°, then AB is a diameter of the circle through A, B and C.
Lines meeting circles. Substitute the line into the circle. A positive discriminant means two points, zero means a tangent, and negative means no points.
Two circles. Compare the distance between the centres with the sum and the difference of the radii, with a sketch.
Tangents from a point. By Pythagoras, the tangent length squared is the distance to the centre squared minus r².
Worked example
Question 1 from this chapter. The points A(−1, 4) and B(5, −4) are the ends of a diameter of the circle C. Find an equation for C.
The centre is the midpoint of AB: ((−1+5)/2, (4−4)/2) = (2, 0).
The radius is the distance from the centre to A: r² = (−1−2)² + (4−0)² = 9 + 16 = 25, so r = 5. (Equivalently, AB = 10 and r is half of that.)
So C has equation (x−2)² + y² = 25.
Seen on real papers
Every question in our booklets is original. These are the recent papers where each question type has appeared.
- June 2023, Q3: the radius from a centre and a point, the circle's equation, then the tangent at the point, as in Q3.
- June 2024, Q7(a): completing the square to find a centre and radius, as in Q2 and Q6.
- June 2024, Q7(b): showing two circles do not meet, where a sketch revealed the simplest method, as in Q6(c).
Where marks are lost
- Signs and square roots. Sign errors in the centre, forgetting to square root, and forgetting to subtract the constants when completing the square were the common slips in June 2024.
- Halving the radius. In June 2023 some students treated the distance from the centre to a point as a diameter and halved it.
- The wrong point for a tangent. Using the centre instead of the point of contact in the tangent's equation was a regular error in June 2023.
- Two circles with no numbers. In June 2024 many compared the distance between the centres with one radius instead of the sum, or gave a conclusion with no values to support it.
Common questions
How do I show two circles do not meet?
Find the distance between the centres and compare it with the sum of the radii (and the difference, in case one is inside the other), giving the values and a conclusion.
How do I find where a line meets a circle?
Substitute the line's equation into the circle's, simplify to a quadratic and solve it, then find the matching y values.
Why draw a sketch?
The June 2024 examiners noted that students who sketched the two circles often saw the far simpler method using the distance between the centres.
Where this chapter leads
- Pure 4 Chapter 3: Coordinate Geometry: parametric equations, including circles written parametrically.
- Pure 4 Chapter 5: Differentiation: tangents to circles found by implicit differentiation.