Focus and Directrix of a Parabola

A parabola is the set of all points in the plane that are equidistant from a fixed point and a straight line. The fixed point is called the focus and the straight line is called the directrix.

Here, we will see more detailed definitions for the focus and the directrix. Then, we will learn about the relationship that these two elements of the parabola have. We will use diagrams to illustrate the concepts.

PRECALCULUS
focus and directrix of a parabola opening upwards

Relevant for

Learning about the focus and the directrix of a parabola.

See definitions

PRECALCULUS
focus and directrix of a parabola opening upwards

Relevant for

Learning about the focus and the directrix of a parabola.

See definitions

Focus of a parabola

The focus of a parabola is the fixed point located inside a parabola that is used in the formal definition of the curve.

A parabola is defined as follows: For a fixed point, called the focus, and a straight line, called the directrix, a parabola is the set of points so that the distance to the focus and to the directrix is the same.

focus of a parabola

The equation of a vertically oriented parabola is $latex {{(x-h)}^2} = 4p(y-k)$. On the other hand, if a parabola is oriented horizontally, its equation is $latex {{(y-k)}^2}=4p(x-h)$. In these equations, p is the distance from the vertex to the focus. Both the vertex and the focus are located on the axis of symmetry.


Directrix of a parabola

The directrix of a parabola is a line perpendicular to the axis of symmetry used in the definition of the parabola. If the axis of symmetry is vertical, the directrix is horizontal.

When the focus is above the directrix, as shown in the following diagram, the parabola opens upwards.

focus and directrix of a parabola opening upwards

When the focus is below the directrix, the parabola opens downwards.

focus and directrix of a parabola opening downwards

Relationship between focus and directrix

The purple lines in the diagram below represent the distance between the focus and the different points on the directrix. Each point on the parabola is equidistant from the focus and the directrix.

relationship between focus and directrix of a parabola

This means that the distance $latex d_{1}$ from the focus to the parabola is the same as the distance $latex d_{1}$ from the directrix to the parabola. The same applies for all other distances from a point on the parabola to the focus and directrix. This is the main characteristic of a parabola.


See also

Interested in learning more about parabolas? Take a look at these pages:

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