If you construct a parabola y=ax^2 using coordinate system and a parabola using focus and directrix.relation?

Question

Wats the relation between the parameter "a" and "the distance between focus and directrix" if the two constructed parabolas superimpose on each other?

Answer 1

A parabola is the set of points equidistant from a line (the directrix) and a point not on the line (the focus).

It is clear that the focus has to be "inside" the parabola and on the axis of symmetry, and that the directrix has to be "outside the parabola and perpendicular to the axis of symmetry.

Looking at y = ax^2, we see
1. the axis of symmetry is the Y axis
2. The point <0,0> is on this parabola.

So lets put the focus (the point defining the parabola) at <0, Y> and the directrix (the line) at y = -Y. This guarantees that the generated parabola includes the point <0,0> and has the right axis of symmetry

So the parabola define by this focus and directrix has the form:

distance to the focus = distance to the directrix
sqrt(x^2 + (Y-y)^2) = Y + y

square both sides to get
x^2 + (Y-y)^2 = (Y+y)^2
x^2 + Y^2 + y^2 - 2yY = Y^2 + 2Yy + y^2

the Y^2 and y^2 terms cancel, and we can get y in terms of Y and x^2. This is supposed to be the same equation as y = ax^2 so you can determine the connection between Y and a.