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 | y^{2} = 4ax |
y^{2} = -4ax |
x^{2} = 4ay |
x^{2} = -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 |

## Video Lesson

### Vertex and Directrix of Parabola

### 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 y^{2} = -12x.

**Solution:**

Given equation of parabola is y^{2} = -12x …(i)

This equation has y^{2} term.

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

Comparing (i) with the equation y^{2} = -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 5y^{2} = 16x, find the vertex, focus and directrix.

**Solution:**

Given equation is 5y^{2} = 16x

y^{2} = (16/5)x

Comparing above equation with y^{2} = 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 x^{2}.

**Solution:**

Given equation is y = ½ x^{2}

Rearranging we get x^{2} = 2y

The axis of the parabola is y-axis.

Comparing the given equation with x^{2} = 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).

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## Frequently Asked Questions

### What do you mean by a Parabola?

Parabola is a locus of a point, which moves so that distance from a fixed point (focus) is equal to the distance from a fixed line (directrix).

### What do you mean by the vertex of a Parabola?

Vertex is the point where the parabola intersects the axis.

### Give the equation of the directrix of the Parabola y^{2}=4ax.

The equation of the directrix of the parabola y^{2}=4ax is given by x = -a.

### What is the focus of the Parabola y^{2}=4ax?

The focus of the Parabola y^{2}=4ax is (a, 0).

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