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Conic Sections

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Introduction (History)

In 300 B.C. Menaechmus, and Euclid studied conic sections just for the beauty of the mathematics. In 200 B.C. Apollonius first used the terms parabola, ellipse and hyperbola. For many years the usefulness of these slices of the cone was unknown. Now, here you are in the year 2004 A.D. studying these same quadratic equations. How do conic sections help us describe and predict natural phenomena? How are we able to apply the properties of these ancient curves in this technological age?

What are Conic Sections?

A conic section is the intersection of a plane and a right circular cone. By changing the angle of the plane the intersection can be: a circle, an ellipse, a parabola, or a hyperbola. If the plane intersects the vertex of the cone the resulting intersection is a point, line, or intersecting lines (these are called degenerate conics). We are mainly interested in the first four with their center (circle, ellipse and hyperbola) or vertex (parabola) located at the origin. Occasionally we’ll be using conics that have been shifted


Parabolas

Circles

Ellipses

Hyperbolas

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