How to get formula for focus and directrix

[ilya.s]

Hello everyone, this is an amazing forum! :)

I have a question: I have seen the formulas for focus/directrix of a parabola, but I cannot trust something unless I know how to get there.

How do I get the formulas of focus and directrix?

I got to an equation (X - Xf)^2 + (Y - Yf)^2 = (Y - Yd)^2 (distance between any point and focus equals to the distance between the same point and directrix). Where do I go from here?

Thank you!

 

[LCKurtz]

You might start with a simple case where the focus and directrix are equidistant from the origin and dirextrix parallel to an axis: Focus (0,p) and directrix y = -p or symmetrically, Focus (p,0), directrix x = -p.

See http://en.wikipedia.org/wiki/Parabola.

 

[BloodyFrozen]

You can derive the formula using the distance formula.