Parabola

A parabola is the open curve formed by the intersection of a plane and a right circular cone. It occurs when the plane is parallel to one of the generatrices of the cone (Figure 1).

A parabola can also be defined as the set of points which are equidistant from a fixed point (the "focus") and a fixed line (the "directrix") (Figure 2).

A third definition is the set of points (x,y) on the coordinate plane which satisfy an equation of the form y = x², or, more usefully, 4ky = x². Other forms of equation are possible, but these are the simplest.

The "axis" of a parabola is the line which passes through the focus and is perpendicular to the directrix. The "vertex" is the point where the axis crosses the parabola. The "latus rectum" is the chord passing through the focus and perpendicular to the axis. Its length is four times the distance from the focus to the vertex.

When a parabola is described by the equation 4ky = x², the vertex is at the origin; the focus is at (o,k); the axis is the y-axis; the directrix is the line y = -k.

In spite of the infinitude of cones—from skinny ones to fat ones—that yield parabolas, all parabolas are geometrically similar. If one has two parabolas, one of them can always be enlarged, as with a photographic enlarger, so that it exactly matches the other. This can be shown algebraically with an example. If y = x² and y = 3x² are two parabolas, the transformation x = 3xy = 3y which enlarges a figure to three times its original size, transforms y = x² into 3y = (3x)², which can be simplified to y = 3x².

This reflects the fact that all parabolas have the same eccentricity, namely 1. The eccentricity of a conic section is the ratio of the distances point-to-focus divided by point-to-directrix, which is the same for all the points on the conic section. Since, for a parabola, these two distances are always equal, their ratio is always 1.

A parabola can be thought of as a kind of limiting shape for an ellipse, as its eccentricity approaches 1. Many of the properties of ellipses are shared, with slight modifications, by parabolas. One such property is the way in which a line intersects it. In the case of an ellipse, any line which intersects it and is not simply tangent to it, intersects it in two points. So, surprisingly, does a line intersecting a parabola, with one exception. A line which is parallel to the parabola's axis will intersect in a single point, but if it misses being parallel by any amount, however small, it will intersect the parabola a second time. The parabola continues to widen as it leaves the vertex, but it does so in this curious way.

A parabola's shape is responsible for another curious property. If one draws a tangent to a parabola at any point P, a line FP from the focus to P and a line XP parallel to the axis, will make equal angles with the tangent. In Figure 3, - FPA =- XPB. This means that a ray of light parallel to the axis of a parabola would be reflected (if the parabola were reflective) through the focus, or a ray of light, originating at the focus, would be reflected along a line parallel to the axis.

A parabola, being an open curve, does not enclose an area. If one draws a chord between two points on the parabola, however, the parabolic segment formed does have an area, and this area is given by a remarkable formula discovered by Archimedes in the third century B.C. In Figure 4, M is the midpoint of the chord AB. C is the point where a line through M and parallel to the axis intersects the parabola. The area of the parabolic segment is 4/3 times the area of triangle ABC. For example, the area of the parabola y = x² and the line y = 9 is (4/3)(6 × 9/2) or 36. What is particularly remarkable about this formula is that it does not involve the number π as the formulas for the areas of circles and ellipses do.