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 y2 = 4px

For the parabola [y2 = 4px],
the focus is (p, 0).

Parabola - Proof of the formula y2 = 4px

Example 1: Focus of y2 = 8x

Change the right side into 4px.

Then 8x = 4⋅2⋅x.

y2 = 4⋅2⋅x

So the focus is (2, 0).

This is the graph of y2 = 4⋅2⋅x.

Its focus is (2, 0).

Example 2: Focus of y2 = -4x

Change the right side into 4px.

Then -4x = 4⋅(-1)⋅x.

y2 = 4⋅(-1)⋅x

So the focus is (-1, 0).

This is the graph of y2 = 4⋅(-1)⋅x.

Its focus is (-1, 0).

Formula: Focus of x2 = 4py

The graph of x2 = 4py looks like this.

For the parabola [x2 = 4py],
the focus is (0, p).

Parabola - Proof of the formula x2 = 4py

Example 3: Focus of y = x2

Switch both sides:
x2 = y.

Then change the right side into 4py.
So y = 4⋅(1/4)⋅y.

x2 = 4⋅(1/4)⋅y

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

This is the graph of x2 = 4⋅(1/4)⋅y.

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