Introduction

Introduction

The Greek mathematicians Archimedes and Apollonius were among the first to study, in real depth, the family of curves now called conic sections — so named because every one of them is exactly what you get by intersecting a plane with a right circular cone.

What you already know. A straight line and a circle are themselves conic sections, and both have already been studied in earlier chapters. This chapter adds three more curves to that list: the parabola, the ellipse, and the hyperbola.

A recap of the section formula, which reappears at points later in this chapter: if $A(x_1,y_1)$ and $B(x_2,y_2)$ are two points in a plane, and a point $P$ divides segment $AB$ internally in the ratio $m:n$, then

and a point $Q$ dividing $AB$ externally in the same ratio $m:n$ is

Why conic sections matter. Far from being purely abstract, conic sections show up constantly in the real world: the parabola in the shape of a flashlight or car-headlight reflector, and in the path of a projectile; the ellipse in the orbits planets trace around the sun; all three curves in the design of telescopes, antennas, and bridges, and in problems of navigation. This chapter builds the tools needed to work with all of them — starting from the double cone that gives conics their name, through a single focus–directrix definition that unifies all three curves, down to the standard equation of each.