Geometry of the Parabola

The parabola is a conic section in which its locus is the set of all points equidistant to a fixed point called the focus and a fixed line called the directrix.

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The parabola is one of the primary conic sections. In any engineering or mathematics application, we'll see a lot of applications:

Describing projectile trajectory
Designing vertical curves in roads and highways
Making reflectors and telescope lenses.

These are a few examples of its many uses. For now, let's learn more about the basic geometry of the parabola:

Locus Definition

At its basic, it is a set of all points that is equidistant to:

A fixed point \(F\) called the focus,
A fixed line called the directrix.

To expand, let's consider a point \((x, y)\) as shown in the figure. The distance \(d_1\) between this point and \(F\) should be equal to its perpendicular length \(d_2\) to the directrix.

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Parts of a Parabola

The parabola has certain notable parts to consider:

Parts of a Parabola

Vertex \(V\) is a point halfway between the focus \(F\) and the directrix. It has the coordinate \((h, k)\).
Focus \(F\) is a fixed point at which \((x, y)\) is equidistant to that of the directrix.
Directrix is a fixed-line at which \((x, y)\) is equidistant to that of the focus.
Directed distance, \(a\) is the halfway distance between the directrix and \(F\).
Axis is the line that passes through \(V\) and \(F\). Depending on the orientation, it may be vertical, horizontal, or inclined. In addition, the graph is symmetrical about this axis.
A focal chord is any line segment that passes through \(F\) and has its endpoints on the parabola.
The Latus Rectum is a focal chord that is perpendicular to the axis. It has a length equal to \(4a\).

Summary

The parabola is a set of all points equidistant to a fixed point \(F\) called the focus and a fixed line called the directrix.

The parabola has certain notable parts: vertex, focus, directrix, directed distance, axis, focal chord, and the latus rectum.

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Orienting the Parabola

Concept

The parabola can be oriented in different ways. The most common orientations are vertical and horizontal parabolas. We'll explore the properties of each in this post.

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