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Mathematics, 17.07.2021 08:40 weeblordd
Where does the 4p come from in (x-a)^2 = 4p(y-k) when dealing with the focus and directrix of a parabola.
Going to use (a, b) as the focus and y = k for the directrix
Mostly going to be talking about vertical parabolas, but I think I can use whatever answers I get to figure out horizontal ones. Everything I'm writing out here I've figured out for both.
Equating the distance between the focus and a point on the graph (namely the vertex) and the vertex and directrix gets me ![\sqrt{(y-k)^2}=\sqrt{(x-a)^2+(y-b)^2}](/tpl/images/2425/1452/b7ede.png)
Then expanding and rearranging I get
Whichh I rearrange to because it's in y=f(x) form and it's basically a complicated vertex form.
Using that information and assuming I have a simple vertex form of a parabola and want to find the focus and directrix, where my Parabola is in the form A(x-C) + D
and 2D = b + k, which is enough to find the focus and directrix since the x coordinate of the focus will be a in
or C in A(x-C)^2 + D (for vertical parabolas anyway)
I've seen the whole (x-a)^2 = 4p(y-k) form though and I'm wondering where that is from, and how it is derived. According to it the focus would be (h, k+p) and the directrix would be y = k-p. Totally expecting it to be simple and something I'm just missing lol
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