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Directrix

A parabola is set of all points in a plane which are an equal distance away from a given point and given line. The point is called the focus of the parabola, and the line is called the directrix .

The directrix is perpendicular to the axis of symmetry of a parabola and does not touch the parabola. If the axis of symmetry of a parabola is vertical, the directrix is a horizontal line .

Math diagram

If we consider only parabolas that open upwards or downwards, then the directrix is a horizontal line of the form $y = c$ .

Relation between focus, vertex and directrix:

The vertex of the parabola is at equal distance between focus and the directrix.

If $F$ is the focus of the parabola, $V$ is the vertex and $D$ is the intersection point of the directrix and the axis of symmetry, then $V$ is the midpoint of the line segment $\bar{F D}$ .

Math diagram

Example:

If a parabola has a vertical axis of symmetry with vertex at $(1, 4)$ and focus at $(1, 2)$ , find the equation of the directrix.

If $F$ is the focus of the parabola, $V$ is the vertex and $D$ is the intersection point of the directrix and the axis of symmetry, then $V$ is the midpoint of the line segment $\bar{F D}$ .

Equate the $x$ -coordinates and solve for $p$ .

$\begin{array}{l} 1 = \frac{1 + p}{2} \\ 2 = 1 + p \\ p = 1 \end{array}$

Equate the $y$ -coordinates and solve for $q$ .

$\begin{array}{l} 4 = \frac{2 + q}{2} \\ 8 = 2 + q \\ q = 6 \end{array}$

The equation of the directrix is of the form $y = c$ and it passes through the point $(1, 6)$ . Here, $c = 6$ .

So, the equation of the directrix is $y = 6$ .

The graph is as shown.

Math diagram