Geometry of the Parabola (2D)

by Henri Picciotto

Parabolas are a central topic in high school algebra classes, but, perhaps because of the rigid separation between algebra and geometry classes in the US secondary curriculum, we do not usually treat them as geometric objects. While most teachers are aware of some of the parabola's geometric properties, few of us are familiar with the proofs of those properties.

On this page, I present the basic geometry of the parabola:
Geometric Definition
Construction
Reflection Property
All Parabolas are Similar
and on the next page:
Conic Section

Geometric Definition

Definition: A parabola is the set of points in the plane that are equidistant from a point (the focus) and a line (the directrix.)

The following exercise should help convince you that this definition yields the parabolas you are familiar with.

Exercise: Given a focus at (0,1) and a directrix y=-1, find the equation of the parabola. How to do it: draw a figure showing a generic point P on the parabola, with coordinates (x,y). Calculate its distance to the focus, its distance to the directrix, set those equal, and simplify. Or, for a more general result, do this exercise for a focus at (0,f) and a directrix y=-f.

Top

Construction

Given the focus (F) and directrix (d), here is a method to construct any number of points on the parabola: choose a point T on d. Construct the perpendicular bisector of TF. Construct the perpendicular to d through T. The intersection of these two lines (P) is a point on the parabola. (Make sure you understand why.)

You can move F and T.
Download the file: GeoGebra | Cabri

Exercise: With the help of dynamic geometry software, construct P as outlined above, then trace P as T moves, or create its locus, which is the parabola.

Low-tech alternative / prequel: Use this focus-directrix graph paper to construct points on a parabola by hand. Select a line to be the directrix, and use the graph paper lines to find points equidistant from it and the focus.
Extra Challenges (using interactive geometry software):
Top

Reflection Property

A light ray originating at the focus will be reflected on the parabola and continue in a direction parallel to the axis of symmetry. Likewise, a light ray coming in parallel to the axis of symmetry will be reflected to hit the focus.

That this works is readily proved using the above construction, if you assume a basic fact from optics: the angle of incidence equals the angle of reflection. The key to the proof is realizing that MP must be tangent to the parabola. Indeed, if it intersected it again at a point P', that point would be equidistant from F and T, but it would necessarily be further from or closer to d, and thus could not be on the parabola -- a contradiction. So P' cannot exist, and MP is a tangent.

Exercise: Prove the reflection property of the parabola, assuming that the angles of incidence and reflection are determined with respect to the tangent to the parabola at the point of incidence.

This property is of course the basis of many applications (headlights, flashlights, satellite dishes, radar...) For example, here is a diagram of how this works in a reflector telescope:

telescope

The primary mirror is parabolic, reflecting the parallel rays to the focus. The secondary (flat) mirror redirects this towards the eyepiece.

Top

All Parabolas are Similar

Like squares and circles, unlike rectangles and ellipses, all parabolas are similar. They cannot be "pointier" or "wider". They all have exactly the same shape, which appears "pointier" from afar, and "wider" when looked at in the neighborhood of the vertex.

Unfortunately, many of us have misled many students by implying otherwise: we often claim that changing the value of a in the formula y=ax2 changes the shape of the parabola. In fact, many teachers believe this to be true. Here are three types of arguments to show it is a misunderstanding.

Algebraic Argument:
y=ax2
ay=a2x2
ay=(ax)2
In other words, in the equation y=x2, both x and y have been multiplied by the same number a. The parabola is scaled with no distortion.

Geometric Argument:
Since the directrix is infinite, moving the focus has no effect on the parabola's shape. It is merely zooming in or out on one shape.

Visual Argument:
Same equation, apparently different shapes:

normal

(Dan Bennett suggests a dramatic illustration of this: make a transparency of a figure like the one above. Project it. Use another transparency to trace a piece of the projection, like the one below. Compare the two transparencies, which seem to have very different shapes, but clearly must represent the same equation. The same approach works on a document camera: place the original next to the projected image of a photocopy of itself.)

close-up

In the illustration below, you can use the Zoom In or Zoom Out tool. Just click near the origin, and see how the same parabola's shape appears to change.

Download the file: GeoGebra | Cabri

Conversely, look at how different equations can yield the same shape:

normal

zoom in

In fact, you can see for yourself: in the applet below, drag the axes' unit (the "1" on the x-axis) left or right, and watch the "a" in the equation change while the parabola's shape remains absolutely constant.

Download the file: GeoGebra | Cabri

Finally, in this illustration, you can show that any two parabolas are similar by translating the first, and then dilating it, so that the final image is superposed on the second. (Use the sliders in order.)

Download the GeoGebra file

Thanks to Kim Seashore for helping me think about "all parabolas are similar."

On to Geometry of the Parabola (3D),
or (easier) Geometry of the Conic Sections (3D).

P icon
Related pages on this site:

Parabolas and Quadratics
Geometry of the Conic Sections (2D)
Geometric Transformations

Top