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# 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.)

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

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.

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.

Now, in order to eliminate the x'y'-term, you must select θ such that B' = 0, as follows.

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

Thus, you have established the desired results.