Parabola definition (focus - directrix form)

Definition: The locus of all points that are equidistant from a given point (focus)
and a given line (directrix).
Try this Drag the point P. It will move through all the points equidistant from the focus point and the directrix line.

Recall that the definition of locus is the set points that meet some given conditions. In this definition of a parabola, it is the shape created by the points that are the same distance from a given point (call the focus) and a given line (called the directrix)*.

Put another way, of all the infinite number of points on the plane, we select only those that are the same distance from the point and the line.

In the figure above, as you drag the point P, the applet only lets it visit the locations that satisfy that condition. As you can see, P is always the same distance from the focus and the directrix*.

* The distance from P to the directrix is always measured as the shortest distance to the line. That is, the perpendicular distance.

The dimensions of the parabola

When defined this way, the "width" of the parabola is determined by the distance between the focus and the directrix. In the figure above, drag the focus point and see the effect it has on the parabola.

Derive the equation of the parabola

When defined this way, we can derive the equation of the parabola. With the origin on the parabola's vertex, the equation is

To see how it derived, see Parabola equation derivation.

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