The ebook Elementary Calculus is based on material originally written by H.J. Keisler. For more information please read the copyright pages.


Second Degree Curves (Cases)

A second degree equation is an equation of the form

(1)

Ax2 + Bxy + Cy2 + Dx + Ey + F = 0.

The graph of such an equation will be a conic section: a parabola, ellipse, hyperbola, or one of several degenerate cases. In Section 5.4 we saw that the graph of a second degree equation of one of the forms

(2)

Ax2 + Dx + Ey + F = 0

or

(3)

Cy2 + Dx + Ey + F = 0

is a parabola or degenerate. In Section 5.5 we saw that the graph of a second degree equation of the form

(4)

Ax2 + Cy2 + F = 0

is an ellipse, a hyperbola, or degenerate.

In this and the next section we shall see how to describe and sketch the graph of any second degree equation. We will begin with the Discriminant Test, which shows at once whether a nondegenerate curve is a parabola, ellipse, or hyperbola. The next topic in this section will be translation of axes, which can change any second degree equation with no xy-term,

(5)

Ax2 + Cy2 + Dx + Ey + F = 0,

into an equation of one of the simple forms (2), (3), or (4).

In the following section we will study rotation of axes, which can change any second degree equation into an equation of the form (5) with no xy-term. We will then be able to deal with any second degree equation by using first rotation and then translation of axes.


Last Update: 2006-11-25