Parabola: Directrix

Parabola: Directrix

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

Formula: Focus of y2 = 4px

For the parabola y^2 = 4px, the directrix of the parabola is x = -p.

For the parabola [y2 = 4px],
the directrix is [x = -p].

Parabola - Proof of the formula y2 = 4px

Example 1: Directrix of y2 = 8x

Find the directrix of the given parabola. y^2 = 8x

Change the right side into 4px.

Then 8x = 4⋅2⋅x.

y2 = 4⋅2⋅x

So the directrix is x = -2.

This is the graph of y2 = 4⋅2⋅x.

Its directrix is x = -2.

Example 2: Directrix of y2 = -4x

Find the directrix of the given parabola. y^2 = -4x

Change the right side into 4px.

Then -4x = 4⋅(-1)⋅x.

y2 = 4⋅(-1)⋅x

So the directrix is x = -(-1).

-(-1) = +1

So x = 1 is the answer.

This is the graph of y2 = 4⋅(-1)⋅x.

Its directrix is x = -(-1).

Formula: Focus of x2 = 4py

For the parabola x^2 = 4py, the directrix of the parabola is y = -p.

For the parabola [x2 = 4py],
the directrix is [y = -p].

Parabola - Proof of the formula x2 = 4py

Example 3: Directrix of y = x2

Find the directrix of the given parabola. y = x^2

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 directrix is y = -1/4.

This is the graph of x2 = 4⋅(1/4)⋅y.

Its directrix is y = -1/4.