subject
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}

Then expanding and rearranging I get y=\frac{1}{2(b-k)}(x-a)^2 +.5(b+k) 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 \frac{1}{2A} = b - k and 2D = b + k, which is enough to find the focus and directrix since the x coordinate of the focus will be a in y=\frac{1}{2(b-k)}(x-a)^2 +.5(b+k) 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

ansver
Answers: 3

Another question on Mathematics

question
Mathematics, 21.06.2019 18:30
Agraphic designer created a logo on eight and a half by 11 inch paper. in order to be placed on a business card, the logo needs to be one and 7/10 inches by 2 and 1/5 in. what is the scale factor of the dilation
Answers: 1
question
Mathematics, 21.06.2019 19:30
If you can solve all of these i will give ! - 4% of 190 - 4% of 162.5 - 4% of 140 - a 4% increase from 155.1 - a 4% increase from 159.8
Answers: 2
question
Mathematics, 21.06.2019 19:50
Asequence is defined recursively using the formula f(n + 1) =f(n) - 5. which sequence could be
Answers: 1
question
Mathematics, 21.06.2019 20:00
Worth 30 points! in this diagram, both polygons are regular. what is the value, in degrees, of the sum of the measures of angles abc and abd?
Answers: 2
You know the right answer?
Where does the 4p come from in (x-a)^2 = 4p(y-k) when dealing with the focus and directrix of a para...
Questions