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GENERAL EQUATION OF THE SECOND DEGREE

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

1) Ax^{2} + 2Bxy + Cy^{2} + 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 b^{2} - ac under rotation. The quantity B^{2} - AC is
invariant under rotation of coordinates. In other words, if the general equation of the second
degree

Ax^{2} + 2Bxy + Cy^{2} + 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

then

B'^{ 2} - A'C' = B^{2} - AC .

● If an equation of the second degree

Ax^{2} + 2Bxy + Cy^{2} + 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' = B^{2} - AC

and we obtain the following result:

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

Ax^{2} + 2Bxy + Cy^{2} + 2Dx + 2Ey + F = 0

can be determined from the value of the invariant quantity B^{2} - AC as follows:

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

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

B |
parabola |

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