Parabola: Equation

How to use a parabola equation to find the focus and the directrix (and vice versa): definition, formula, 6 examples, and their solutions.

Definition

Definition

A parabola is the set of the points
that are equidistant
from the focus (blue point)
and the directrix (dashed line).

Formula: y2 = 4px

Formula

The black curve is the graph of a parabola
y2 = 4px.

For y2 = 4px,
the focus is (p, 0)
and the directrix is x = -p.

Example 1: Focus

Example

Solution

Change the parabola to y2 = 4px form.

y2 = 4⋅2⋅x

y2 = 4⋅2⋅x

Then the focus is
(2, 0).

So (2, 0) is the answer.

Example 2: Directrix

Example

Solution

You just found that
the given equation y2 = 8x is
y2 = 4⋅2⋅x.

y2 = 4⋅2⋅x

Then the directrix is
x = -2.

So x = -2 is the answer.

Example 3: Equation

Example

Solution

The focus is (3, 0).
And the directrix is x = -3.

Then the equation of the parabola is
y2 = 4⋅3⋅x.

4⋅3⋅x = 12x

So y2 = 12x is the answer.

Formula: x2 = 4py

Formula

For the parabola x2 = 4py,
the focus is (0, p)
and the directrix is y = -p.

Example 4: Focus

Example

Solution

Change the parabola to x2 = 4py form.

Switch both sides.

y = 4⋅[1/4]⋅y

x2 = 4⋅[1/4]⋅y

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

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

Example 5: Directrix

Example

Solution

You just found that
the given equation y = x2 is
x2 = 4⋅[1/4]⋅y.

x2 = 4⋅[1/4]⋅y

Then the directrix is
y = -1/4.

So y = -1/4 is the answer.

Example 6: Equation

Example

Solution

The focus is (0, 2).
And the directrix is y = -2.

Then the equation of the parabola is
x2 = 4⋅2⋅y.

4⋅2⋅y = 8y

So x2 = 8y is the answer.