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Suppose we went to find an equation for an ellipse whose axes of symmetry are not horizontal or vertical, but are tilted with respect to the x and y-axes.

In general, the computations are quite involved so we content ourselves with working out one example in detail.

EXAMPLE
Consider the ellipse E1 shown in Figure 1. Its major axis has length 4 and lies along the line y = x; its minor axis has length 2 and its centre is at the origin.
Figure 1
We can start with a congruent ellipse with centre at the origin and major axis along the x-axis as in Figure 2.
Figure 2
In standard form, the equation of the ellipse E2 is:  x2 + y2 = 1 22
In general form, this equation becomes: x2 + 4y2 - 4 = 0
Now we want to rotate E2 by 45o in a counter-clockwise direction to get E1. Any point (x, y) on E2 moves to a new point (x', y') on E1, where x' = (x - y) and Ö2 y' = (x + y)        Ö2
Check out a derivation of these marvelous relations.
To find out what equation the point (x', y') satisfies, you need to convert the x and y in x2 + 4y2 - 4 = 0 (the equation of E1) into x' and y'. So you need expressions of x' and y'. This is easy: x' = (x - y)       (1) Ö2     y' = (x + y)        (2) Ö2      Add (1) and (2): x' + y' = 2x = Ö2 x   Ö2 Thus x = (x' + y')        (3) Ö2      Subtract (1) from (2) y' - x' = 2y = Ö2y     Ö2 Thus y = (y' - x')        (4) Ö2
Now substitute (3) and (4) into the equation for E2: x2 + 4y2 - 4 = 0 ==> [(x'+ y')/ Ö2 ]2 + 4[(y' - x')/ Ö2 ]2 - 4 = 0 ==> (x'2 + 2x'y' + y'2)/ 2 + 2(y'2 - 2x'y' + x'2) - 4 = 0 ==> x'2 + 2x'y' + y'2 + 4y'2 - 8x'y' + 4x'2 - 8 = 0 ==> 5x'2 + 5y'2 - 6x'y' - 8 = 0
Now, (x', y') was just an arbitrary point on E2, so the variable notation can be changed. Thus E1 is the set of all points (x, y) such that 5x2 + 5y2 - 6xy - 8 = 0.

The action figure below allows you to change A, B, C, D, E and F to show the graph of Ax2 + By2 + Cx + Dy + Exy + F = 0. The ellipse E1 appears or disappears when you click on the blue button. Create a red graph of the equation 5x2 + 5y2 - 6xy - 8 = 0 to check our work. It is most interesting to watch if you use the arrows to click F to -8 first, followed by E to -6, A to 5 and then B to 5.

If all was done correctly, the red ellipse should cover the blue ellipse.