The General Quadratic Equation in Two Variables
Part I - Rotations

A second order equation in two variables is any equation that, when simplified (parentheses removed, all terms moved to the left side of the equation, and like terms combined), is equivalent to an equation of the form ...

Ax² + Bxy + Cy² + Dx + Ey + F = 0

... where A, B, C, D, E, and F are real constants.

Rotation of Axis

In order to simplify the study of these equations, rotation of axis must first be considered. If the coordinate system is rotated about the origin through and angle of θ (measured in degrees or radians), how will an equation change relative to this new coordinate system? As seen in the image to the right, the red point has coordinates (x, y) relative to the black coordinate system. The blue coordinate system is obtained by rotating the black coordinate system θ degrees (or radians) in the counterclockwise direction. That same red point will have coordinates (x^*, y^*) relative to the blue coordinate system.

This leads to the question: What is the relationship between (x, y) and (x^*, y^*)? To answer this question, draw a line from the origin to the point and let α be the angle between this line and the x^* axis (as seen in the diagram). Also, let r be the distance from the origin to the point. Trigonometry tells us that for the ordered pair of the red point relative to the black coordinate system, (x, y) ...

x = r cos(θ+α)
y = r sin(θ+α)

or

x = r cosθ cosα - r sinθ sin α
y = r sinθ cosα + r cosθ sin α

For the ordered pair of the red point relative to the blue coordinate system, (x^*, y^*) ...

x^* = r cosα
y^* = r sinα

Substituting these into the previous result (as highlighted in blue) gives ...

x = x^*cosθ - y^*sinθ
y = x^*sinθ + y^*cosθ

This last pair of equations can be used to convert an equation in terms of the xy-coordinate system into an equation in terms of the x^*y^*-coordinate system. For example, consider the linear equation, 2x + 3y = 6 (a line with x-intercept of 3 and a y-intercept of 2). What would be the equation of this same line if the coordinate system was rotated 30° in the counterclockwise direction (without moving the line)? Using the above pair of equations with θ = 30° ...

Substituting these into the original equation of the line gives ...

2x + 3y = 6

Therefore, the line crosses the x^*-axis at approximately 1.86 and the y^*-axis at approximately 3.75 (as shown in the following figure).

Eliminating the XY term from the General Quadratic Equation: Ax² + Bxy + Cy² + Dx + Ey + F = 0

To "simplify" this equation, substitute the rotation of axis identities into this equation. That is, let x = x^*cosθ - y^*sinθ and y = x^*sinθ + y^*cosθ. This gives ...

A(x^*cosθ - y^*sinθ)² + B(x^*cosθ - y^*sinθ)(x^*sinθ + y^*cosθ) + C(x^*sinθ + y^*cosθ)² + D(x^*cosθ - y^*sinθ) + E(x^*sinθ + y^*cosθ) + F = 0

Multiplying everything out and combining like terms gives ...

[A cos²θ + B cosθ sinθ + C sin²θ]x^*2 + [B cos2θ - (A-C) sin2θ]x^*y^* + [A sin²θ - B cosθ sinθ + C cos²θ]y^*2 +
[D cosθ + E sinθ ]x^* + [-D sinθ + E cosθ]y^* + F = 0

The desired result is to choose an angle θ such that the coefficient of the x^*y^* term is zero (eliminating this term from the equation). That is,

B cos2θ - (A-C) sin2θ = 0

B cos2θ = (A-C) sin2θ

[Note: In practice, the second form of this last equations is used because calculators do not include the cotangent function or its inverse. However, if A = C, the second form is undefined and therefore the first form must be used. In this case, the angle is equal to 45° or π/4 radians.]

Summary

An equation of the form Ax² + Bxy + Cy² + Dx + Ey + F = 0, where B ≠ 0, can be changed to an equation of the form ...

A^*x^*2 + C^*y^*2 + D^*x^* + E^*y^* + F^* = 0

... by letting ...

x = x^*cosθ - y^*sinθ
y = x^*sinθ + y^*cosθ

... with ...

The new coefficients are given by the equations ...

A* = A cos2θ + B cosθ sinθ + C sin2θ C* = A sin2θ - B cosθ sinθ + C cos2θ D* = D cosθ + E sinθ E* = -D sinθ + E cosθ F* = F

The following Rotation Calculator uses the above formulas to determine the new coefficients of an equation after a rotation that will eliminate the Bxy term (i.e. B^* = 0). Calculations are rounded to the nearest 100th of a unit.

Rotation Calculator
A =	A* =
B =	B* = 0
C =	C* =
D =	D* =
E =	E* =
F =	F* =
	θ = Degrees

Therefore, this procedure reduces the analysis of the general quadratic equation in two variables to equations of the form ...

Ax² + Cy² + Dx + Ey + F = 0

If the original equation includes a Bxy term, the above transformation will be applied to give an equation without a term involving both variables. After this new equation is graphed, it will be rotated in a counterclockwise direction about the origin through the angle θ to get the graph of the original equation.

The Discriminant

The expression B² - 4AC is often referred to as the discriminant. For any quadratic equation in two variables, the value of this expression does not change when rotations of axis is applied. Hence, it is called an invariant. We can show that this is true using the above relationships, that is:

A^* = A cos²θ + B cosθ sinθ + C sin²θ

B^* = B cos2θ - (A-C) sin2θ = B cos²θ - B sin²θ - 2A cosθ sinθ + 2C cosθ sinθ

C^* = A sin²θ - B cosθ sinθ + C cos²θ

So, (B*)2 - 4A*C*	= (B cos2θ - B sin2θ - 2A cosθ sinθ + 2C cosθ sinθ)2 - 4(A cos2θ + B cosθ sinθ + C sin2θ)(A sin2θ - B cosθ sinθ + C cos2θ)
	= B2 cos4θ - 2B2 cos2θ sin2θ - 4AB cos3θ sinθ + 4BC cos3θ sinθ + B2 sin4θ + 4AB cosθ sin3θ - 4BC cosθ sin3θ + 4A2 cos2θ sin2θ - 8AC cos2θ sin2θ + 4C2 cos2θ sin2θ- 4A2 cos2θ sin2θ + 4AB cos3θ sinθ - 4AC cos4θ - 4AB cosθ sin3θ + 4B2 cos2θ sin2θ - 4BC cos3θ sinθ - 4AC sin4θ + 4BC cosθ sin3θ - 4C2 cos2θ sin2θ
	= (B2 - 4AC) cos4θ + (B2 - 4AC) sin4θ + 2(B2 - 4AC) cos2θ sin2θ
	= (B2 - 4AC)(cos4θ + 2 cos2θ sin2θ + sin4θ)
	= (B2 - 4AC)(cos2θ + sin2θ)2
	= B2 - 4AC

This fact allows us to determine the type of conic that is determined by a quadratic equation that includes a Bxy term without eliminating the Bxy term. Later we will see that an equation without a Bxy term is a parabola if A or C is zero (i.e. AC = 0), an ellipse if A and C have the same sign (i.e. AC > 0), or a hyperbola if A and C have opposite signs (i.e. AC < 0). Therefore, since the discriminant is an invariant under rotations of axis, if ...

• B² - 4AC = 0, the graph of the quadratic equation is a parabola.
• B² - 4AC < 0, the graph of the quadratic equation is an ellipse.
• B² - 4AC > 0, the graph of the quadratic equation is a hyperbola.

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Copyright © 2007 by Ron Wallace, all rights reserved