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As part of my portfolio for my end of semester math classes, I am trying to develop the equation for a parabola at an arbitrary rotation. I have seen it done based off of substituting X and Y variables with trig terms, and I am confident I could prove it that way, but I am curious if I can do it another way. When I took analytic geometry, we derived the equation for a parabola starting with an equation that simply equated the distance from a point to a line (directrix) and the distance from a point to a another specific point (the focus) and from there we derived the equation of a parabola. What I am trying to do is simply rewrite that equality in a form that holds true regardless of whether or not the directrix (and the parabola) is rotated or not. I have already figured out (I hope so at least) that the distance from an arbitrary point to the directrix where the line is perpendicular to the directrix should be represented by: [tex]\frac{y-tan(\theta)x(sin(90-\theta))}{sin(90)}[/tex]

Where θ is the angle of rotation.

Now I think that all I should have to do is equate that to a formula describing the distance from a point to the focus and then simplify, but I am having trouble coming up with the second part.

Can anyone point me on the right track, or show me why my reasoning so far is wrong?

Thank you,