Analytic Geometry

The Greek mathematician Menaechmus (c. 380–c. 320 BCE) is generally credited with discovering the shapes formed by the intersection of a plane and a right circular cone: tilt the plane differently and a different curve appears, each with near-perfect symmetry. Aristotle presumed the planets moved in circular orbits around Earth, and for nearly 2000 years that was the common belief; not until Johannes Kepler published his laws of planetary motion in the 1600s did the orbits turn out to be ovals with the sun at one focus. Other objects follow similar elliptical paths — including the rings of Saturn, which 19th-century mathematicians such as James Clerk Maxwell and Sofya Kovalevskaya showed are not solid discs but billions of small particles, a structure whose full understanding still relies on mathematical analysis. In this chapter, we investigate the two-dimensional figures formed when a right circular cone is intersected by a plane, develop defining equations for each, and use those equations to solve a variety of problems.

Sections

• The Ellipse — write equations of ellipses in standard form from vertices, foci, and general-form equations; graph ellipses centered at the origin and at $(h,k)$; and solve applied problems such as whispering chambers and elliptical arches.
• The Hyperbola — locate a hyperbola’s vertices and foci; write its equation in standard form; graph hyperbolas centered at the origin and at $(h,k)$ with their asymptotes and central rectangle; and solve applied problems modeled by hyperbolas.
• The Parabola — graph parabolas with vertices at the origin and at $(h,k)$ from the focus, directrix, and latus rectum; write equations of parabolas in the standard form $(x-h)^2=4p(y-k)$; and solve applied problems involving parabolic reflectors and arches.
• Rotation of Axes — identify nondegenerate conic sections from their general-form equations, use the rotation-of-axes formulas to eliminate an $xy$ term, write rotated conics in standard form, and identify a conic from its discriminant without rotating axes.
• Conic Sections in Polar Coordinates — identify a conic from its polar equation by its eccentricity, graph the polar equations of conics, and define conics in terms of a focus and a directrix.

The sections above are adapted from Precalculus 2e, Chapter 10 by Jay Abramson, © OpenStax, licensed under CC BY-NC-SA 4.0. Access the original for free at openstax.org. The chapter opener’s Cassini photograph of Saturn’s rings is omitted; its historical framing is condensed above.