Extensions Page 4b

Page 4b

Locus Definition of an Parabola

Unlike ellipses and hyperbolas, parabolas have only one focal point.
Instead of another focal point, each parabola has a fixed line called a directrix which performs a special function.

Using the focal point and the directrix, you can define any parabola:
A parabola is the locus of all points whose perpendicular distance from a fixed line (the directrix) is the same as the distance from a fixed point (the focal point).
In the figures above, F represents the focal point of the parabola, and D represents the point on the directrix where a perpendicular line from some point P on the parabola hits the directrix.
Notice that PF represents the distance between P on the parabola and the focal point F, and PD represents the distance between P on the parabola and the point D on the directrix.
Then PF = PD no matter where P is located on the parabola. This is shown in the applet below. Use the arrow buttons to move the blue point labelled F. When it is on top of the black dot, it is located at the "focal point" and you can see that PF = PD by looking at the graph on the right.

Unlike ellipses and hyperbolas, parabolas have only one focal point.
Instead of another focal point, each parabola has a fixed line called a directrix which performs a special function.

Using the focal point and the directrix, you can define any parabola: A parabola is the locus of all points whose perpendicular distance from a fixed line (the directrix) is the same as the distance from a fixed point (the focal point).
In the figures above, F represents the focal point of the parabola, and D represents the point on the directrix where a perpendicular line from some point P on the parabola hits the directrix.
Notice that PF represents the distance between P on the parabola and the focal point F, and PD represents the distance between P on the parabola and the point D on the directrix.
Then PF = PD no matter where P is located on the parabola. This is shown in the applet below. Use the arrow buttons to move the blue point labelled F. When it is on top of the black dot, it is located at the "focal point" and you can see that PF = PD by looking at the graph on the right.