Equation of a Tangent to a Circle

Overview

Finding the equation of a tangent to a circle is a core skill in OCR's Further Mathematics specification (3.2). This topic elegantly combines coordinate geometry with the geometric properties of circles, specifically the crucial relationship between a tangent and a radius. In the exam, candidates are expected to confidently handle circles with centers at the origin and elsewhere, and to be proficient in manipulating linear equations. A typical question will not only test your ability to find the equation but also your precision in presenting it in the specific form (ax + by + c = 0) with integer coefficients. Mastering this methodical process is key to unlocking a significant number of marks and demonstrating a deeper understanding of geometric principles.

Key Concepts

Concept 1: The Tangent-Radius Property

The single most important concept is that a tangent to a circle is perpendicular to the radius at the point of contact. This is a fundamental circle theorem that forms the basis of the entire method. When two lines are perpendicular, their gradients (let's call them (m_1) and (m_2)) have a special relationship: their product is -1. This is the mathematical key to solving these problems.

Relationship: (m_{radius}\times m_{tangent} = -1)

This means if you can find the gradient of the radius connecting the circle's center to the point of contact, you can immediately find the gradient of the tangent by calculating the negative reciprocal.

Concept 2: The Methodical Approach

Every question on this topic can be solved by following a clear, five-step process. Committing this process to memory ensures you won't miss any crucial steps, especially under exam pressure. Examiners look for this logical progression, and marks are awarded for each stage of the process.

Mathematical/Scientific Relationships

• Gradient of a Line: Given two points ((x_1, y_1)) and ((x_2, y_2)), the gradient (m) is (m =\frac{y_2 - y_1}{x_2 - x_1}). (Must memorise)
• Perpendicular Gradients: For two perpendicular lines with gradients (m_1) and (m_2), (m_1 \times m_2 = -1). This can be rearranged to (m_2 = -\frac{1}{m_1}). (Must memorise)
• Equation of a Straight Line (Point-Slope Form): Given a point ((x_1, y_1)) and a gradient (m), the equation is (y - y_1 = m(x - x_1)). (Given on formula sheet)
• Standard Form of a Linear Equation: (ax + by + c = 0), where a, b, and c are integers. (Must memorise form)

Practical Applications

While abstract, the concept of tangents to circles has real-world applications in fields like physics (describing the instantaneous velocity of an object in circular motion), computer graphics (calculating light reflections and shadows), and engineering (designing gears and pulley systems where belts run tangent to wheels).