by rotating the coordinate axes through an angle
θ, where
Note that the constant term is the same in both equations.
Because of this, F is said to be invariant under rotation. The following
theorem lists some
other rotation invariants. The proof of this theorem is left as an exercise.
Theorem 2
Rotation Invariants
The rotation of coordinate axes through an angle
θ that transforms the equation Ax2 + Bxy + Cy2 + Dx + Ey + F = 0
into the form
A'(x')2 + B'x'y' + C'(y')2 + D'x' + E'y' + F' = 0
has the following rotation invariants.
1. F = F'
2. A + C = A' + C'
3. B2 - 4AC = (B')2 - 4A'C'
You can use this theorem to classify the graph of a second-degree equation with
an xy-term in much the same way you do for a second-degree equation without an xy-term.
Note that because B' = 0, the invariant B2 - 4AC reduces to
B2 - 4AC = - 4A'C'
Discriminant
which is called the discriminant of the equation
Ax2 + Bxy + Cy2 + Dx + Ey + F = 0
Because the sign of A'C' determines the type of graph for the equation
A'(x')2 + B'x'y' + C'(y')2 + D'x' + E'y' + F' = 0
the sign of B2 - 4AC must determine the type of graph for the original equation. This
result is stated in the following theorem.
Theorem 3
Classification of Conics by the Discriminant
Ax2 + Bxy + Cy2 + Dx + Ey + F = 0
is, except in degenerate cases, determined by its discriminant as follows.
1. Ellipse or circle
B2 - 4AC < 0
2. Parabola
B2 - 4AC = 0
3. Hyperbola
B2 - 4AC > 0
Example
Using the Discriminant
Classify the graph of each of the following equations.
