Directrix of a Parabola

Definition

The directrix of a parabola is a fixed straight line not touching or intersecting the parabola such that the distance of any point on the parabola from directrix is equal to its distance from the focus.

The directrix is always perpendicular to the axis of the parabola and lies on the side opposite to which the parabola is facing (the parabola is said to be facing on the side where it opens - for example, the parabola below is right facing).

In the following graph of the parabola `y^2 = 4x`, the directrix is `x = -1` and focus is labelled S. A, B, and C are three points on the parabola. The distance of each point from the focus equals its distance from the directrix. This means that AP = AS, BR = BS, and CQ = CS.

Directrix of a Parabola

For right or left facing parabolas, like the one above, the directrix is a vertical line. For upright or down facing parabolas, their directrices are horizontal lines (Directrices is plural for directrix).

An interesting property of the directrix is that the vertex is the mid point of the line joining the focus and the directrix. This can be understood from the fact that the distance between any point on the parabola and the directrix is equal to the distance between that point and the focus. The vertex is a point on the parabola, hence its distance from the directrix is equal to its distance from the focus. Since both the focus and vertex lie on the same straight line (the axis), hence the vertex is the midpoint of the focus and the point of intersection of the directrix and the axis. Thus, knowing the coordinates of any two, say the vertex and focus, the coordinates of the third point can be be obtained  (the point of intersection in this case) by using the midpoint formula.

The equation of the directrix of a parabola can be determined by following a few simple steps. These steps are explained in this post.

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Axis of a parabola