How To Find The Vertex Focus and Directrix Of The Parabola With The Given Equation

A parabola is a plane curve formed by a point moving so that its distance from a fixed point is equal to its distance from a fixed-line. The fixed-line is the directrix of the parabola and the fixed point is the focus denoted by F. The axis of the parabola is the line through the F and perpendicular to the directrix. The point where the parabola intersects the axis is called the vertex of the parabola. In this article, we will learn how to find the vertex focus and directrix of the parabola with the given equation.

Steps to Find Vertex Focus and Directrix Of The Parabola

Step 1. Determine the horizontal or vertical axis of symmetry.

Step 2. Write the standard equation.

Step 3. Compare the given equation with the standard equation and find the value of a.

Step 4. Find the focus, vertex and directrix using the equations given in the following table.

The following table gives the equation for vertex, focus and directrix of the parabola with the given equation.

Equation of parabola y2 = 4ax y2 = -4ax x2 = 4ay x2 = -4ay
Vertex (0,0) (0,0) (0,0) (0,0)
Focus (a,0) (-a,0) (0,a) (0, -a)
Equation of directrix x = -a x = a y = -a y = a

Solved Examples

Let us have a look at some examples.

Example 1:

Find the vertex, focus, the equation of directrix and length of the latus rectum of the parabola y2 = -12x.

Solution:

Given equation of parabola is y2 = -12x …(i)

This equation has y2 term.

So the axis of the parabola is the x-axis.

Comparing (i) with the equation y2 = -4ax

We can write -12x = -4ax

So a = 12/4 = 3

Focus is (-a,0) = (-3,0).

Equation of directrix is x = a.

I.e x = 3 is the required equation for directrix.

Vertex is (0,0).

Length of latus rectum = 4a = 4×3 = 12.

Example 2.

Given the equation of a parabola 5y2 = 16x, find the vertex, focus and directrix.

Solution:

Given equation is 5y2 = 16x

y2 = (16/5)x

Comparing above equation with y2 = 4ax

We get, 4a = 16/5

a = ⅘.

Focus is (a,0) = (4/5,0).

Equation of directrix is x = -a.

I.e x = -⅘

Vertex is (0,0).

Example 3.

How to find the directrix, focus and vertex of a parabola y = ½ x2.

Solution:

Given equation is y = ½ x2

Rearranging we get x2 = 2y

The axis of the parabola is y-axis.

Comparing the given equation with x2 = 4ay

We get 4ay = 2y

a = 2/4 = ½

Focus is (0,a) = (0, ½ )

Equation of directrix is y = -a.

I.e y = -½ is the equation of directrix.

Vertex of the parabola is (0,0).