Become a math whiz with AI Tutoring, Practice Questions & more.

Join for Free
HotmathMath Homework. Do It Faster, Learn It Better.

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 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 F D ¯ .

Equate the x -coordinates and solve for p .

1 = 1 + p 2 2 = 1 + p p = 1

Equate the y -coordinates and solve for q .

4 = 2 + q 2 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.

Math diagram

 

Subjects Near Me
Popular Cities
Popular Subjects
;
Download our free learning tools apps and test prep books
varsity tutors app storevarsity tutors google play storevarsity tutors amazon storevarsity tutors ibooks store