Rotating Axes
You probably know already that equations of conics with axes parallel to one of the
coordinate axes can be written in the general form
Ax2 + Cy2 + Dx + Ey + F = 0 |
Horizontal or vertical axes |
Here you will study the equations of conics whose axes are rotated so that they are not
parallel to the x-axis or the y-axis. The general equation for such conics contains an
xy-term.
Ax2 + Bxy + Cy2 + Dx + Ey + F = 0 |
Equation in xy-plane
|
To eliminate this xy-term, you can use a procedure called rotation of axes. You want
to rotate the x- and y-axes until they are parallel to the axes of the conic. (The rotated
axes are denoted as the x'-axis and the y'-axis, as shown in the figure below.)
![](./articles_imgs/206/algebr1.gif)
After the
rotation has been accomplished, the equation of the conic in the new x'y'-plane will
have the form
A'(x')2 + B'x'y' + C'(y')2 + D'x' +
E'y' + F' = 0 |
Equation in x'y'-plane
|
Because this equation has no x'y'-term, you can obtain a standard form by completing
the square.
The following theorem identifies how much to rotate the axes to eliminate an
xy-term and also the equations for determining the new coefficients A', C', D',
E',
and F'.
THEOREM 1
Rotation of Axes
The general equation of the conic
Ax2 + Bxy + Cy2 + Dx + Ey + F = 0
where B ≠ 0, can be rewritten as
A'(x')2 + B'x'y' + C'(y')2 + D'x' + E'y' + F' = 0
by rotating the coordinate axes through an angle
θ, where
![](./articles_imgs/206/algebr2.gif)
The coefficients of the new equation are obtained by making the substitutions
x = x' cos θ - y' sin θ
y = x' sin θ + y' cos θ.
Proof
To discover how the coordinates in the xy-system are related to the coordinates
in the x'y'-system, choose a point P = (x, y) in the original system and attempt to find
its coordinates (x', y') in the rotated system. In either system, the distance r between
the point P and the origin is the same, and thus the equations for x, y, x', and
y' are
those given in the figure below.
![](./articles_imgs/206/algebr3.gif) ![](./articles_imgs/206/algebr4.gif)
Using the formulas for the sine and cosine of the difference
of two angles, you obtain
x' = r cos (α -
θ) |
= r (cos α cos θ
+ sin α sin θ) = r cos α cos θ + r
sin α sin θ = x cos θ + y sin
θ |
y' = r sin (α -
θ) |
= r (sin α cos θ
- cos α sin θ) = r sin α cos θ - r
cos α sin θ = y cos θ - x sin
θ |
Solving this system for x and y yields
x = x' cos θ - y' sin θ and y = x' sin
θ + y' cos θ.
Finally, by substituting these values for x and y into the original equation and collecting
terms, you obtain the following.
![](./articles_imgs/206/algebr8.gif)
Now, in order to eliminate the x'y'-term, you must select
θ such that B' = 0, as
follows.
![](./articles_imgs/206/algebr7.gif)
If B = 0, no rotation is necessary, because the xy-term is not present in the original
equation. If B ≠ 0, the only way to make B'
= 0 is to let
![](./articles_imgs/206/algebr6.gif)
Thus, you have established the desired results.
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