Invariants Under Rotation
Have a look at the following theorem:
THEOREM 1
Rotation of Axes
The general equation of the conic
Ax^{2} + Bxy + Cy^{2} + Dx + Ey + F = 0
where B ≠ 0, can be rewritten as
A'(x')^{2} + B'x'y' + C'(y')^{2} + D'x' + E'y' + F' = 0
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 Ax^{2} + Bxy + Cy^{2} + 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. B^{2}  4AC = (B')^{2}  4A'C'
You can use this theorem to classify the graph of a seconddegree equation with
an xyterm in much the same way you do for a seconddegree equation without an xyterm.
Note that because B' = 0, the invariant B^{2}  4AC reduces to
B^{2}  4AC =  4A'C' 
Discriminant 
which is called the discriminant of the equation
Ax^{2} + Bxy + Cy^{2} + 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 B^{2}  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
Ax^{2} + Bxy + Cy^{2} + Dx + Ey + F = 0
is, except in degenerate cases, determined by its discriminant as follows.
1. Ellipse or circle 
B^{2}  4AC < 0 
2. Parabola 
B^{2}  4AC = 0 
3. Hyperbola 
B^{2}  4AC > 0 
Example
Using the Discriminant
Classify the graph of each of the following equations.
a. 4xy  9 = 0
b. 2x^{2}  3xy + 2y^{2}  2x = 0
c. x^{2}  6xy + 9y^{2}  2y + 1 = 0
d. 3x^{2} + 8xy + 4y^{2}  7 = 0
Solution
a. The graph is a hyperbola because
B^{2}  4AC = 16  0 > 0.
b. The graph is a circle or an ellipse because
B^{2}  4AC = 9  16 < 0.
c. The graph is a parabola because
B^{2}  4AC = 36  36 = 0
d. The graph is a hyperbola because
B^{2}  4AC = 64  48 > 0
