Introduction

From Circles to a New Family of Curves

The previous two chapters studied a single circle, and then several circles together (system of circles). This chapter turns to a different curve entirely: the parabola. Its name comes from the Greek mathematician Apollonius of Perga (c. 262–190 BCE), and quite fittingly, describes the same U-shaped path traced by a ball thrown through the air.

What This Chapter Covers

A parabola is one of the four conic sections — the curves (circle, parabola, ellipse, hyperbola) obtained by slicing a double cone with a plane at different angles. This chapter develops the algebraic definition that makes conics easy to compute with: fixing a point (the focus) and a line (the directrix), a conic is the locus of every point whose distance from the focus, divided by its distance from the directrix, equals a fixed constant $e$ (the eccentricity). A parabola is exactly the case $e=1$ — later chapters show that $0<e<1$ gives an ellipse and $e>1$ gives a hyperbola. From this locus definition, the chapter derives the parabola's standard equation $y^2=4ax$, its different standard forms, and the equations of its tangent and normal at a point.