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Course: College Algebra
Topic: Conic Sections
Subtopic: General and Degenerate Conics

Overview

In calculus you will be expected to know how to identify and graph conics from both the general form of a conic, ax2+bx+cy2+dy+e=0, and the standard form. The key to understanding the relationship between these two forms of a conic's equation and the graph is the process of converting from one to the other including via completing the square.

Sometimes, depending on the a b c d e constants, the equation will result in a point, line, or two intersecting lines. These cases are called the degenerate conics and occur when the plane slicing the double cone does so at particularly interesting places or at specific angles. These degenerates provide a great connection between the algebra of conics and the visual experience. Cool stuff!

Objectives

By the end of this topic you should know and be prepared to be tested on:

• 7.5.1 Understand the general equation of a conic section
• 7.5.2 Algebraically convert from the general form and standard form of a conic and visa versa using CTS as needed
• 7.5.3 Identify a conic from the general form equation
• 7.5.4 Understand degenerate conics including their equations, graphs, formation (in terms of slicing the double cone), and how to identify them from the general equation of a conic

Terminology

Define: general form of the equation of a conic section, degenerate conic

Supplemental Resources

Download/Print: Conic Sections Formula Sheet (by Prof. Louise Hoover, ret. Clark College)