Sheet № 200 · Higher only · AQA · Edexcel · OCR
Equation of a Circle –
The equation of a circle is a Higher tier GCSE Maths topic that connects algebra and geometry. You need to recognise the standard form, find the radius, and solve problems involving tangents to a circle at a given point.
§Key definitions
Question:
This is a Higher only topic. Here is a basic entry. A circle has the equation x² + y² = 36. State the centre and radius.
Answer:
Centre (0, 0), radius 6.
Q1 (Higher):
A circle has equation x² + y² = 100. State the radius.
Q2 (Higher):
Does the point (1, 7) lie on the circle x² + y² = 50?
Q3 (Higher):
Find the equation of the tangent to x² + y² = 20 at (4, 2).
§Formulas to memorise
x² + y² = r² — circle centred at the origin with radius r
Gradient of radius × gradient of tangent = -1
To find the radius, take the square root of the constant: r = √(right-hand side).
To check if a point lies on the circle, substitute its x and y values — if x² + y² equals r², the point is on the circle.
The tangent gradient is the negative reciprocal. Use y - y₁ = m(x - x₁) to write the tangent equation.
Worked example
This is a Higher only topic. Here is a basic entry. A circle has the equation x² + y² = 36. State the centre and radius.
Working:
⚠ Common mistakes
- ✗Confusing r and r². If x² + y² = 49, the radius is 7 (not 49). Always take the square root.
- ✗Forgetting the tangent is perpendicular to the radius. Students sometimes use the radius gradient as the tangent gradient instead of the negative reciprocal.
- ✗Assuming the centre is not at the origin. At GCSE, the equation x² + y² = r² always has its centre at (0, 0). If the equation includes (x - a)² + (y - b)² = r², the centre is (a, b) — but this is rare at GCSE.
✦ Exam tips
- →If you see x² + y² = a number, immediately think "circle, centre origin, radius = square root."
- →To find where a line intersects a circle, substitute the equation of the line into the circle equation and solve the resulting quadratic.
- →For tangent questions, always start by confirming the point lies on the circle.