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:
Figure 5.1: The x and y axes are rotated through an angle of θ to their
new position as the x' and y' axes.
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:
| 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.
