The Focus-Directrix Definition of a Parabola
Simple Explanation
A parabola is the set of all points equidistant from a fixed point (the focus) and a fixed line (the directrix) — every point on the curve is exactly as far from the focus as it is from the directrix.
Why Do We Need It?
This geometric definition is what actually generates the parabola's curve — the algebraic equations covered next are derived directly from this defining property.
See It
A parabola opening upward, with a point marked as its focus above the vertex, and a horizontal dashed line below the vertex representing the directrix
Formula
The Focus-Directrix Definition of a Parabola
distance to focus = distance to directrix, for every point on the parabola
A parabola is the set of all points equidistant from a fixed point (the focus) and a fixed line (the directrix).
- focus
- — a fixed point inside the curve of the parabola
- directrix
- — a fixed line on the outside of the curve, the same perpendicular distance rule applies to
When to use it: Whenever a parabola needs to be defined or verified from its geometric (rather than algebraic) definition.
Worked Example
Verify a point lies on a parabola using the focus-directrix definition
A parabola has focus (0,2) and directrix y=−2. Verify that the point (4,2) lies on this parabola.
Why Does This Work?
This equidistance condition is literally the DEFINITION of a parabola — any point satisfying it is, by definition, on the curve, and any point on the curve must satisfy it; this is exactly how the standard algebraic equation (covered next) gets derived.
Real-Life Example
Satellite dish and headlight reflector design
A satellite dish or car headlight uses a parabolic reflector, taking advantage of a special reflective property tied directly to the focus.
All incoming signals parallel to the axis reflect toward the single focus point (or, in a headlight, light from the focus reflects out in parallel rays) — a direct physical consequence of the focus-directrix definition.
Practice
A parabola has focus (0,3) and directrix y=−3. Find the distance from the point (6,3) to the focus.
MediumCommon mistake
Computing the distance to the directrix incorrectly — remember it is the PERPENDICULAR distance to the line itself, not the distance to some specific point on it.
Quick Review
- A parabola: every point is equidistant from the focus and the directrix.
- This defining property is what the algebraic equation is built from.
- The reflective property of parabolas (used in dishes and headlights) follows directly from this definition.