Development of a Parabola from the Definition

A parabola is the set of all points in a plane that are equidistant between a fixed point (focus) and a line (directrix).

In its simplest form, the parabola with focal length p has its vertex at the origin (0,0) and the focus is at the point (0,p). The directrix is the line y=-p.

Any point (x,y) on the parabola will be the same distance from the focus as it is from the directrix. That is, if d₁ is the distance from the focus to the point on the parabola, and d₂ is the distance from the directrix to the point on the parabola, then d₁=d₂.

Since d₁ is the distance between two points, (0,p) and (x,y), we need to use the distance formula to find it.

d₂ is the distance between the point (x,y) and a horizontal line y=-p. The x-coordinates will be the same, so the distance between the point and line is the difference in the y-values.

We earlier said that the parabola is where d₁ = d₂. Let's set them equal to each other and then square both sides to get rid of the square root.

Now, let's expand both sides

Move everything but the x² to the right side and simplify. This gives you the standard form for a parabola with vertex at the origin and opening up.

x² = 4py

Return to notes on conic sections.