Equation of a parabola - derivation

Given a parabola with focal length f, we can derive the equation of the parabola. (see figure on right). We assume the origin (0,0) of the coordinate system is at the parabola's vertex.

For any point (x,y) on the parabola, the two blue lines labelled d have the same length, because this is the definition of a parabola. So we can find an equation for each of them, set them equal to each other and simplify to find the parabola's equation.

The top line d is the hypotenuse of the small right triangle. The horizontal side of the triangle has a length of x. The vertical side has a length of (y–f).

From the Pythagorean theorem: Turning to the vertical line d. From Parabola definition (focus-directrix) we know the vertex is always half way between the focus and the directrix, so: So we now have two equations for d. We know they are equal, so we can set the equations equal to each other: We now simplify this, aiming to get y on the left side. So, first square both sides to get rid of the radical: Expanding the squared expressions The y2 and f2 terms on each side cancel: Add 2fy to each side Divide both sides by 4f

If the origin is on the directrix

We could have chosen to have the origin of the coordinate system on the directrix. In this case the coordinate grid would move down by an amount equal to f. So we have to 'move' the parabola up* by the same amount by adding f to the equation:

Alternatively, you could repeat the above, but replacing f with 2f everywhere.

*  To see how adding f moves the parabola up, see Quadratic explorer.

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