# Parabola: Focus

How to find the focus of the given parabola: definition, formulas, examples, and their solutions.

## Definition

A parabola is the set of the points

that are equidistant

from the focus and the directrix.

## Formula: Focus of *y*^{2} = 4*px*

For the parabola [*y*^{2} = 4*px*],

the focus is (*p*, 0).

Parabola - Proof of the formula *y*^{2} = 4*px*

## Example 1: Focus of *y*^{2} = 8*x*

Change the right side into 4*px*.

Then 8*x* = 4⋅2⋅*x*.

*y*^{2} = 4⋅2⋅*x*

So the focus is (2, 0).

This is the graph of *y*^{2} = 4⋅2⋅*x*.

Its focus is (2, 0).

## Example 2: Focus of *y*^{2} = -4*x*

Change the right side into 4*px*.

Then -4*x* = 4⋅(-1)⋅*x*.

*y*^{2} = 4⋅(-1)⋅*x*

So the focus is (-1, 0).

This is the graph of *y*^{2} = 4⋅(-1)⋅*x*.

Its focus is (-1, 0).

## Formula: Focus of *x*^{2} = 4*py*

The graph of *x*^{2} = 4*py* looks like this.

For the parabola [*x*^{2} = 4*py*],

the focus is (0, *p*).

Parabola - Proof of the formula *x*^{2} = 4*py*

## Example 3: Focus of *y* = *x*^{2}

Switch both sides:*x*^{2} = *y*.

Then change the right side into 4*py*.

So *y* = 4⋅(1/4)⋅*y*.

*x*^{2} = 4⋅(1/4)⋅*y*

So the focus is (0, 1/4).

This is the graph of *x*^{2} = 4⋅(1/4)⋅*y*.

Its focus is the blue point (0, 1/4).