Chapter 5 Conic Sections

Objectives

After completing this chapter, you should be able to:

Understand the parabola, ellipse, and hyperbola as curves defined by distance conditions relative to foci and a directrix, rather than merely as graphs of particular equations.
Recognize how a single geometric idea—a locus of points satisfying a distance condition—unifies the three conic sections.
Interpret the roles of the focus, directrix, vertices, and axes in determining the shape and orientation of each conic.
Understand the reflection properties of the conics and why they explain applications such as parabolic reflectors and whispering galleries.
Appreciate how the defining property of the hyperbola underlies position-finding methods such as hyperbolic navigation.

Conic sections are the curves obtained by intersecting a double cone with a plane. There are three types: parabolas, ellipses, and hyperbolas. In this section we give the focus–directrix and focal definitions of each, read the key features off the standard equations, and see a worked example together with a second example that changes the orientation.

Instructions.

The surface is the double cone \(x^2 + y^2 = z^2\text{,}\) cut by the plane \(z = m\,y + c\text{.}\) Use the buttons to jump to the three cases, or drag the “tilt” slider yourself and watch the red curve change from an ellipse (\(m \lt 1\text{,}\) plane shallower than the slant) to a parabola (\(m = 1\text{,}\) plane parallel to the slant) to a hyperbola (\(m \gt 1\text{,}\) plane steep enough to cut both nappes). The “offset” slider moves the plane up and down; try setting \(c = 0\) so the plane passes through the apex and the section degenerates to a point, one line, or a pair of crossing lines. Drag the figure to view it from a different angle, or press the “Rotate” button to spin it automatically.

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