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:
- 6.5.1 Understand the general equation of a conic section
- 6.5.2 Algebraically convert from the general form and standard form of a conic and visa versa
- 6.5.3 Identify a conic from the general form equation
- 6.5.4 Understand degenerate conics including their equations, graphs, formation (in terms of slicing the double cone), and how to identify them
Terminology
Define: general form of the equation of a conic section, degenerate conic
Text Notes
Text:
College Algebra 5ed by Blitzer, sect. 7.3 (cont'd)
- Throughout chapter 7 the text works with both the general form of a conic and the standard form of each conic as well as converting from general form to standard form via CTS. But the text neglects to adequately discuss how to identify a conic from the general equation, so let's discuss this in class.
- Page 660 discusses, far too briefly, the "degenerate conics". Since the text neglects to discuss how to identify a degenerate conic from the general equation, let's discuss this in class.