### Introduction

The
**
conic sections
**
(or
**
conics
**
) - the
**
ellipse
**
, the
**
parabola
**
and the
**
hyperbola
**
- play an important role both in mathematics and in the application of mathematics to engineering. In this Section we look in detail at the equations of the conics in both standard form and general form.

Although there are various ways that can be used to define a conic, we concentrate in this Section on defining conics using Cartesian coordinates $\left(x,y\right)$ . However, at the end of this Section we examine an alternative way to obtain the conics.

#### Prerequisites

- be able to factorise simple algebraic expressions
- be able to change the subject in simple algebraic equations
- be able to complete the square in quadratic expressions

#### Learning Outcomes

- understand how conics are obtained as curves of intersection of a double-cone with a plane
- state the standard form of the equations of the ellipse, the parabola and the hyperbola
- classify quadratic expressions in $x,\phantom{\rule{1em}{0ex}}y$ in terms of conics

#### Contents

1 The ellipse, parabola and hyperbola1.1 The ellipse

1.2 The circle

2 Engineering Example 1

2.1 A circle-cutting machine

3 Engineering Example 2

3.1 The web-flange junction

3.2 The parabola

3.3 General conics