Identifying Nondegenerate Conics in General Form
In previous sections of this chapter, we have focused on the standard form equations for nondegenerate conic sections. In this section, we will shift our focus to the general form equation, which can be used for any conic. The general form is set equal to zero, and the terms and coefficients are given in a particular order, as shown below.
where , and are not all zero. We can use the values of the coefficients to identify which type conic is represented by a given equation.
You may notice that the general form equation has an term that we have not seen in any of the standard form equations. As we will discuss later, the term rotates the conic whenever is not equal to zero.
Conic Sections | Example |
---|---|
ellipse | |
circle | |
hyperbola | |
parabola | |
one line | |
intersecting lines | |
parallel lines | |
a point | |
no graph |
A General Note: General Form of Conic Sections
A nondegenerate conic section has the general form
where , and are not all zero.
The table below summarizes the different conic sections where , and and are nonzero real numbers. This indicates that the conic has not been rotated.
ellipse | |
circle | |
hyperbola | , where and are positive |
parabola |
How To: Given the equation of a conic, identify the type of conic.
- Rewrite the equation in the general form, .
- Identify the values of and from the general form.
- If and are nonzero, have the same sign, and are not equal to each other, then the graph is an ellipse.
- If and are equal and nonzero and have the same sign, then the graph is a circle.
- If and are nonzero and have opposite signs, then the graph is a hyperbola.
- If either or is zero, then the graph is a parabola.
Example 1: Identifying a Conic from Its General Form
Identify the graph of each of the following nondegenerate conic sections.Solution
- Rewriting the general form, we have
and , so we observe that and have opposite signs. The graph of this equation is a hyperbola.
- Rewriting the general form, we have
and . We can determine that the equation is a parabola, since is zero.
- Rewriting the general form, we have
and . Because , the graph of this equation is a circle.
- Rewriting the general form, we have
and . Because and , the graph of this equation is an ellipse.
Try It 1
Identify the graph of each of the following nondegenerate conic sections.Licenses & Attributions
CC licensed content, Specific attribution
- Precalculus. Provided by: OpenStax Authored by: OpenStax College. Located at: https://cnx.org/contents/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1/Preface. License: CC BY: Attribution.