CIE IGCSE Additional Maths
Coordinate Geometry of the Circle
Every circle is just a distance equation in disguise — once you know a circle's centre and radius, you can find where lines touch it, cross it, or miss it completely, using nothing but algebra you already know (Pythagoras, gradients, and simultaneous equations).
Summary — What This Chapter Covers
Equation of a circle
Centre & radius from any form
Line meets circle (0, 1, or 2 times)
Tangents to circles
Two circles intersecting
Common chord equation
- A circle with centre
(a, b) and radius r has equation (x−a)² + (y−b)² = r². This comes straight from the distance formula.
- Circles can also appear in general form
x² + y² + 2gx + 2fy + c = 0 — you convert this back to the centre-radius form by completing the square.
- To find where a line meets a circle, substitute the line into the circle equation, simplify to a quadratic, and count the solutions: 2 solutions = crosses twice, 1 (repeated) solution = tangent, 0 real solutions = misses entirely.
- A tangent is always perpendicular to the radius at the point of contact — this is the single most important fact in this whole chapter.
- Two circles can intersect twice, touch once, or not intersect at all — this is decided purely by comparing the distance between centres to the sum and difference of the two radii.
- The common chord (the line joining two intersection points of two circles) can be found either by subtracting the circle equations, or from the two intersection points directly.
1. The Equation of a Circle
Where does the formula even come from?
Picture a circle with its centre sat at the point (a, b), and radius r. Now pick any point (x, y) that lies exactly on the circle's edge. By definition, that point is always a distance r away from the centre — no matter where on the circle it sits.
Recall the distance formula between two points: distance = √[(x₂−x₁)² + (y₂−y₁)²]. Applying that between our point (x, y) and the centre (a, b), and setting the distance equal to r, then squaring both sides to remove the square root, gives you the circle equation. It's not a rule to memorise blindly — it's Pythagoras' theorem wearing a disguise.
y
^
|
| (x, y) <- any point on circle
| /|
| / |
| r |
| / |
| *----+
| (a,b)
+----------------> x
distance from (a,b) to (x,y) is always r
=> (x-a)^2 + (y-b)^2 = r^2
Core Formula
A circle with centre (a, b) and radius r has equation:
(x − a)² + (y − b)² = r²
Reading the equation instantly
Whatever number is being subtracted from x inside the bracket is a (the x-coordinate of the centre) — and whatever is subtracted from y is b. The number on the right-hand side is r², not r — always square-root it to get the actual radius.
Practice Question 1
Write down the equation of the circle with centre (−3, 2) and radius 7.
Practice Question 2
A circle has centre (−5, −7) and passes through the point (1, 1). Find the equation of the circle.
Finding the centre and radius from a given equation
This is the reverse process. Given an equation already in the form (x − a)² + (y − b)² = r²:
- The radius is the positive square root of whatever number sits on the right-hand side.
- centre
x
y