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GENERAL EQUATION OF THE SECOND DEGREE, CONICS, REDUCTION TO CANONICAL FORM, THE 9 CANONICAL FORMS, TRANSFORMATION OF COORDINATES

General equation of the second degree. The general equation of the second degree in two variables is

1)        Ax² + 2Bxy + Cy² + 2Dx + 2Ey + F = 0.

This equation has as its locus a conic. See figure 1. The conics consist of ellipses, hyperbolas, parabolas, and certain limiting or degenerate forms of these (a point, a straight line or a pair of straight lines).

Simplification by a change of coordinate system. The algebraic expression for any curve or surface is dependent on the location and orientation of the coordinate system. When the coordinate system is rotated or moved in any way the expression for the curve or surface changes. The general equation of the second degree can be simplified greatly by a change to a different coordinate system. A suitable rotation of the coordinate system will eliminate the mixed term xy. A suitable translation will eliminate one and possibly both terms in x and y. By a suitable rotation and translation of the coordinate system the general second degree equation can be reduced to one of the following canonical forms:

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TRANSFORMATION OF COORDINATES

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To transform an equation of a curve from an old system of rectangular coordinates (x, y) to a new system of rectangular coordinates (x', y'), substitute for each old variable in the equation of the curve its expression in terms of the new variables.

Translation of coordinate system. Suppose the origin of the new x'-y' system is at point (h, k) of the old x-y system (with the axes of the new system parallel to the corresponding axes of the old system). Then

            x = x' + h

            y = y' + k

Rotation of the axes about the origin. Suppose the origin of the new x'-y' system is coincident with the origin of the old system and the new system is rotated by angle θ from the old system. See figure 3. Then

            x = x' cos θ - y' sin θ

            y = x' sin θ + y' cos θ

Translation and rotation. Suppose the new x'-y' system origin is at point (h, k) of the old system and the new system is rotated by an angle θ from the old system. See figure 4. Then 

            x = x' cos θ - y' sin θ + h

            y = x' sin θ + y' cos θ + k

Example. Suppose we are given the equation

1)        3x² + 4xy + 5y² + 3x + 1 = 0

and wish to know what the equation of this curve would be in an x'-y' system whose origin is at point (h, k) of the old system and where the new x'-y' system is rotated by an angle θ from the old system. To find the equation of the curve in this new x'-y' system we simply substitute in 1) the expressions for x and y in terms of x' and y' as given by  

            x = x' cos θ - y' sin θ + h

            y = x' sin θ + y' cos θ + k

Expanding and simplifying gives the equation of the curve in the x'-y' system.