So far, we have only studied conics whose axes were parallel to the coordinate axes. Such conics can be written with the equation Ax2 + Cy2 + Dx + Ey + F = 0. The coefficient of the xy term, B, equals zero when the conic's axes are parallel to the coordinate axes. When the axes of the conic are not parallel to the coordinate axes, we can still express the conic without using an xy term by rotating the coordinate axes to from a new plane. Instead of using the xy-plane, the conic can be expressed as a second degree equation (Ax2 + Cy2 + Dx + Ey + F = 0) in the x'y'-plane. The new second degree equation of the conic, after the axes have been rotated, will look like this: A'x'2 + C'y'2 + D'x' + E'y' + F' = 0. The advantage to doing this is that by avoiding an xy-term, we can still express the equation of the conic in standard form.
When a conic contains an xy term, the x and y axes can be rotated through an angle of θ such that they are once again parallel with the axes of the conic, thus eliminating the xy term of the conic. A rotation of the coordinate axes looks something like this: The newly rotated axes are called the x' and y' axes.
A point in the x'y'-plane can be expressed with x and y coordinates, and vice-versa, using the following conversion formulas:
|x = x'cos(θ) - y'sin(θ)|
|y = x'sin(θ) + y'cos(θ)|
|x' = x cos(θ) + y sin(θ)|
|y' = - x sin(θ) + y cos(θ)|
Here's an example problem: what are the x and y coordinates of the point (x', y') = (1, 0) in the x'y'-plane which is the xy-plane rotated an angle of radians? Using the formulas above, x = 1 cos() - 0 sin() = , and y = 1 sin() + 0 sin() = . So the x and y coordinates of this point are (,).
To eliminate the xy term of a conic of the form Ax2 + Bxy + Cx2 + Dx + Ey + F = 0 in order to use its standard form and write it in an equation of the form A'x'2 + C'y'2 + D'x' + E'y' + F' = 0, you must rotate the coordinate axes through an angle θ such that cot(2θ) = . Once this is done, and θ is found, you can use the conversion equations above to change the x and y coordinates to x' and y' coordinates. Then standard form can be used and the foci and vertices will be easily recognizable--but they will be in the x' and y' coordinate system. When they are found, it is possible to convert back to x and y coordinates to sketch the conic.