M2: Coordinate Geometry and Circles

Year 1 / ASYear 2 / A-Level All Boards (AQA, Edexcel, OCR, WJEC, CCEA) AQA

Equations of circles, tangents and normals, intersection of lines and circles.

Exam Board:

📋 Key Definitions and Core Concepts

Circle Equation: (x - a)² + (y - b)² = r² with centre (a, b) and radius r.

Tangent: A straight line perpendicular to the radius at the point of contact.

General circle form: x² + y² + 2gx + 2fy + c = 0 has centre (-g, -f) and radius √(g² + f² - c).
Gradient of normal at (x₁, y₁) is (y₁ - b)/(x₁ - a); gradient of tangent is perpendicular negative reciprocal.
To find intersections with line y = mx + c, substitute into circle equation and test discriminant Δ.

Exam-Style Question

Question:

Find the equation of the tangent to (x - 3)² + (y + 2)² = 25 at (7, 1).

Model Solution & Mark Scheme:

Centre (3, -2). m_radius = (1 - (-2))/(7 - 3) = 3/4.
m_tangent = -4/3.
Equation: y - 1 = -4/3(x - 7) => 4x + 3y - 31 = 0.

Q1: Find the centre and radius of x² + y² - 6x + 8y - 11 = 0.

Show Model Answer

Answer: (x - 3)² + (y + 4)² = 36. Centre (3, -4), radius = 6.

Q2: Determine intersections of y = 2x + 5 with x² + y² = 5.

Show Model Answer

Answer: x² + (2x + 5)² = 5 => 5x² + 20x + 20 = 0 => (x + 2)² = 0. Tangent at (-2, 1).

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