The general second-degree equation in two variables appears as:

where A, B, and C are not all 0. (This ensures that there is at least one quadratic term.)

The graphs of this equation is a conic section. All the different conic sections may be obtained by slicing a right circular cone by a plane. These curves are called: a circle, an ellipse, a parabola, and a hyperbola. Which curve you get depends upon the angle you slice the cone and, for the equation, depend upon the values of the coefficients A, B, C, D, E, and F.

An important and useful result is to identify which curve is represented by the equation then to sketch the graph of the equation. This is not difficult since there are some simple general rules.

If the coefficient B is not 0, then a rotation of the axes can be performed to make it 0. That is, a non-zero B means we could produce a simpler situation that is merely rotated with B = 0. The transformation is straight-forward and involves trigonometry. For our purposes here, we will consider only those equations for which indeed B = 0. Therefore, the general second-degree equation becomes:

where A and C are not both 0.

We have two primary situations:

A and C are both not equal to zero

In this case the D and E are related to the position of the curve relative to the coordinate axes. They can be eliminated by completing the square (twice) to obtain:
A(x−h)² + C(y−k)² + F' = 0

where h and k are the new "center" for the curve.
Which curve and its shape is determined by A and C with the size being determine by F'.

A and C have the same sign

A = C:   the result is a circle with radius given by r² = −F'/A

A ≠ C:   the result is an ellipse


To obtain the size and orientation, write the equation in standard form (for h = k = 0):

	x2	+	y2	= 1
	a2		b2

The length along the x-axis is 2a and the length along the y-axis is 2b.



A and C are opposite in sign
the result is a hyperbola
To obtain the size and orientation, write the equation in standard form (for h = k = 0):

x2  −  y2  = 1 a2 b2

or

−  x2  +  y2  = 1 a2 b2

The asymptotes are the diagonals of the box formed x = ±a and y = ±b. The hyperbola is oriented so that it crosses the x-axis if the x² term is positive and the y-axis if the y² term is positive.

A or C (but not both) is equal to zero

the result is a parabola

A = 0  [D ≠ 0]
The general equation becomes (after completing the square to eliminate E):

  C(y−k)² + Dx + F" = 0

this is the equation of a parabola that opens to the left or right



C = 0   [E ≠ 0]
The general equation becomes (after completing the square to eliminate D):

  A(x−h)² + Ey + F" = 0

this is the equation of a parabola that opens up or down