How to derive formula for focus of a parabola?

Question

I understand how to obtain the formula for the vertex of a formula, $ y= a(x-h) + k $ where $ h=-b/2a$ and the vertex is $(h,k)$. However I don't know how to get to $(h,k+1/4a)$. Could someone please explain in a relatively simple way?

Answer 1

A parabola is the locus of points that are equidistant from a point (the focus) and a line (the directrix).

Let the focus be $(0,p)$ and the directrix be $y=-p$. Then, to find the points $(x,y)$ that are equidistant from $(0,p)$ and $y=-p$, we have $$ \overbrace{(x-0)^2+(y-p)^2}^{\text{square of distance from $(0,p)$}} =\overbrace{\ \ (y+p)^2\ \ }^{\text{square of distance from $y=-p$}}\tag{1} $$ Cancelling common terms in $(1)$ we get $$ x^2=4py\tag{2} $$ Translating $(2)$ to $(h,k)$, we get $$ (x-h)^2=4p(y-k)\tag{3} $$ If $a=\frac1{4p}$, then $(3)$ becomes $$ a(x-h)^2+k=y\tag{4} $$ Accounting for the translation applied in equation $(3)$, we get that the focus $(0,p)$ becomes $$ (h,k+p)=\left(h,k+\frac1{4a}\right)\tag{5} $$