Conic sections and parametric equations

Slice a cone with a flat plane, and the edge of the cut is a circle, an ellipse, a parabola, or a hyperbola. These curves are the conic sections. They describe the orbits of planets, the shape of satellite dishes, and the paths of thrown balls.

This chapter finds their equations, turns tilted conics straight, writes all three in one polar equation, and ends with parametric equations, a way to describe motion along a curve.

This chapter covers the following topics:

Parabolas

The parabola as the set of points equally far from a focus and a directrix, its standard equations, shifted parabolas, and the reflecting property.

Ellipses

The ellipse as the set of points whose distances to two foci add up to a constant, its standard equations, eccentricity, and shifted ellipses.

Hyperbolas

The hyperbola as the set of points whose distances to two foci differ by a constant, its asymptotes, and shifted hyperbolas.

Rotation of axes

Equations with an ${xy}$ term, turning the axes to remove it, and identifying a conic from its equation.

Polar equations of conics

One equation for all three conics using eccentricity, with an application to the orbits of planets and comets.

Parametric equations

Describing a curve by giving ${x}$ and ${y}$ in terms of a third variable, eliminating the parameter, projectile motion, and the cycloid.

Practice questions

Practice problems on every section of this chapter, with full step-by-step solutions.