Rotating conic sections, such as ellipses, circles, parabolas, or hyperbolas, is most easily done by rotating the axis on which they lie instead. The angle at which the axis is rotated is represented by theta (θ).
The angle θ can be determined by using the following equation:
cot(2θ) = (A – C) / B
The values of x and y can be determined by the following equations:
x = x'cosθ – y'sinθ
y = x'sinθ + y'cosθ
After substituting the value of θ into these equations, the x and y values are determined, and can be substituted into the original equation. Doing so and simplifying should eliminate the xy term, resulting in an equation that can be used to graph the ellipse, circle, parabola, or hyperbola on the rotated axis.
The rotated axis can be created by drawing a set of x and y axes at the angle θ counterclockwise from the original set of x and y axes.
Normally, this process is very complicated at first, especially at the step where you must simplify. If there is still an xy term after you simplify, this is an easy method of seeing that you have done something wrong. If there is not this xy term, simplifying further should make the equation much smaller and easier to read.
Finally, all you have to do is draw the ellipse, circle, parabola, or hyperbola on the rotated axis just as you would draw it on a regular axis!
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