Determining distance from Focus to Point on Parabola

[Vorde]

Hi,

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,

 

[LCKurtz]

Hi,

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.

You haven't shown how you got that, but unfortunately it can't be correct. Here's why. How much a line has been rotated is not enough to write its equation. After all, saying how much a line is rotated is equivalent to just giving its slope. Given the slope, you don't know the equation of the line until you also know a point on the line. Your formula above doesn't use any information except the rotation. Lots of lines have that rotation and the distance from your point (x,y) to the line can't be the same for all of them.