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

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

is a conic or limiting form of a conic.

● Any equation of the second degree in x and y that contains a term in xy can be transformed by a suitably chosen rotation into an equation that contains no term in xy.

● The rotation angle that will eliminate the xy term is given by


where the rotation transformation equations are

            x = x' cos θ - y' sin θ

            y = x' sin θ + y' cos θ .

Invariance of the quantity b2 - ac under rotation. The quantity B2 - AC is invariant under rotation of coordinates. In other words, if the general equation of the second degree

            Ax2 + 2Bxy + Cy2 + 2Dx + 2Ey + F = 0

is transformed by

            x = x' cos θ - y' sin θ

            y = x' sin θ + y' cos θ

to obtain

            A'x' 2 + 2B'x'y' + C'y'2 + 2D'x' + 2E'y' + F ' = 0


            B' 2 - A'C' = B2 - AC .

● If an equation of the second degree

            Ax2 + 2Bxy + Cy2 + 2Dx + 2Ey + F = 0

is transformed so as to eliminate the xy term to give

            A'x' 2 + C'y'2 + 2D'x' + 2E'y' + F ' = 0

then the conic represented by the equation will be

1) an ellipse (or limiting form of ellipse: a circle, point or imaginary locus) if A' and C ' have the same sign i.e. if A'C' > 0

2) a hyperbola (or limiting form of hyperbola: intersecting straight lines) if A' and C ' have opposite signs i.e. if A'C' < 0

3) a parabola if A' = 0 or C ' = 0 i.e. if A'C' = 0

Because, B' = 0, we have

            - A'C' = B2 - AC

and we obtain the following result:

● The type of conic represented by the general equation of the second degree in two variables

            Ax2 + 2Bxy + Cy2 + 2Dx + 2Ey + F = 0

can be determined from the value of the invariant quantity B2 - AC as follows:

B2 - AC < 0

ellipse or limiting form (i.e. circle, point, imaginary locus)

B2 - AC > 0

hyperbola or limiting form (i.e. intersecting lines)

B2 - AC = 0


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