Geometry of the Parabola (2D)

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.

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.)

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. [The figure above was created in Cabri, where a conic section can be defined by five points. You can drag F or T. To replay the construction step by step, double-click it and use the toolbar that appears at the bottom.]

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. This property is of course the basis of many applications (headlights, telescopes, satellite dishes, ...)

This 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.

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:

(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.)

Different equations, identical shape:

In fact, you can see for yourself: in the Cabri applet below, drag the axes' unit (initially "0.2") left or right on the x-axis, and watch the "a" in the equation change while the parabola's shape remains absolutely constant.

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