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.

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 .

Example 1:

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 .

Let (p, q) be the point D on the directrix and use the midpoint formula.

Equate the x-coordinates and solve for p.

2 = 1 + p

p = 1

Equate the y -coordinates and solve for q .

8 = 2 + q

q = 6

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.