The parabola: focus, directrix and standard form

Covers AS91573

The focus–directrix definition

A parabola is the set of points that are equidistant from a fixed point and a fixed line.
The fixed point is the focus; the fixed line is the directrix.
For any point $P$ on the curve, the distance to the focus equals the perpendicular distance to the directrix.

A parabola with focus, directrix and a point on the curve

Standard form opening right

A parabola with vertex at the origin opening to the right has equation:

$y^2 = 4ax$

$a$ — the distance from the vertex to the focus (and from the vertex to the directrix).
The focus is at $(a, 0)$ .
The directrix is the vertical line $x = -a$ .
The axis of symmetry is the $x$ -axis, $y = 0$ .

The four orientations

Right: $y^2 = 4ax$ — focus $(a, 0)$ , directrix $x = -a$ .
Left: $y^2 = -4ax$ — focus $(-a, 0)$ , directrix $x = a$ .
Up: $x^2 = 4ay$ — focus $(0, a)$ , directrix $y = -a$ .
Down: $x^2 = -4ay$ — focus $(0, -a)$ , directrix $y = a$ .
Which variable is squared tells you the axis: $y^2$ opens sideways, $x^2$ opens up or down.

Finding $a$ , the focus and the directrix

Match the equation to $y^2 = 4ax$ (or the relevant form).
Solve $4a = (\text{coefficient})$ to find $a$ .
Use $a$ to write down the focus and directrix.

Worked ExampleFocus and directrix of a parabola

A parabola has equation $y^2 = 20x$ . Find $a$ , the coordinates of the focus, and the equation of the directrix.

Step 1 — Match to the standard form

Compare $y^2 = 20x$ with $y^2 = 4ax$ , so $4a = 20$ .

Step 2 — Solve for $a$

$a = \frac{20}{4} = 5$

Step 3 — Write down the focus and directrix

The curve opens right, so the focus is at $(a, 0)$ and the directrix is $x = -a$ :

$\boxed{a = 5, \quad \text{focus } (5, 0), \quad \text{directrix } x = -5}$

Test yourself

Practice by grade

One question each at Achieved, Merit and Excellence. Have a go, then compare with the model answer.

Achieved

For the parabola $y^2 = 8x$ , state the coordinates of the focus and the equation of the directrix.

Merit

A satellite dish has a parabolic cross-section with its vertex at the origin, opening upward. The receiver sits at the focus, $0.5$ m above the vertex. Find the equation of the cross-section, and the width of the dish at the height of the receiver.

Excellence

Using the focus–directrix definition, show that the set of points equidistant from the focus $(a, 0)$ and the directrix $x = -a$ satisfies $y^2 = 4ax$ .

The parabola: focus, directrix and standard form

Covers AS91573

The focus–directrix definition

A parabola is the set of points that are equidistant from a fixed point and a fixed line.
The fixed point is the focus; the fixed line is the directrix.
For any point $P$ on the curve, the distance to the focus equals the perpendicular distance to the directrix.

A parabola with focus, directrix and a point on the curve

Standard form opening right

A parabola with vertex at the origin opening to the right has equation:

$y^2 = 4ax$

$a$ — the distance from the vertex to the focus (and from the vertex to the directrix).
The focus is at $(a, 0)$ .
The directrix is the vertical line $x = -a$ .
The axis of symmetry is the $x$ -axis, $y = 0$ .

The four orientations

Right: $y^2 = 4ax$ — focus $(a, 0)$ , directrix $x = -a$ .
Left: $y^2 = -4ax$ — focus $(-a, 0)$ , directrix $x = a$ .
Up: $x^2 = 4ay$ — focus $(0, a)$ , directrix $y = -a$ .
Down: $x^2 = -4ay$ — focus $(0, -a)$ , directrix $y = a$ .
Which variable is squared tells you the axis: $y^2$ opens sideways, $x^2$ opens up or down.

Finding $a$ , the focus and the directrix

Match the equation to $y^2 = 4ax$ (or the relevant form).
Solve $4a = (\text{coefficient})$ to find $a$ .
Use $a$ to write down the focus and directrix.

Worked ExampleFocus and directrix of a parabola

A parabola has equation $y^2 = 20x$ . Find $a$ , the coordinates of the focus, and the equation of the directrix.

Step 1 — Match to the standard form

Compare $y^2 = 20x$ with $y^2 = 4ax$ , so $4a = 20$ .

Step 2 — Solve for $a$

$a = \frac{20}{4} = 5$

Step 3 — Write down the focus and directrix

The curve opens right, so the focus is at $(a, 0)$ and the directrix is $x = -a$ :

$\boxed{a = 5, \quad \text{focus } (5, 0), \quad \text{directrix } x = -5}$

Test yourself

Practice by grade

One question each at Achieved, Merit and Excellence. Have a go, then compare with the model answer.

Achieved

For the parabola $y^2 = 8x$ , state the coordinates of the focus and the equation of the directrix.

Merit

Excellence

Using the focus–directrix definition, show that the set of points equidistant from the focus $(a, 0)$ and the directrix $x = -a$ satisfies $y^2 = 4ax$ .

The parabola: focus, directrix and standard form

The focus–directrix definition

Standard form opening right

The four orientations

Finding aaa, the focus and the directrix

Step 1 — Match to the standard form

Step 2 — Solve for aaa

Step 3 — Write down the focus and directrix

Test yourself

Practice by grade

The parabola: focus, directrix and standard form

The focus–directrix definition

Standard form opening right

The four orientations

Finding aaa, the focus and the directrix

Step 1 — Match to the standard form

Step 2 — Solve for aaa

Step 3 — Write down the focus and directrix

Test yourself

Practice by grade

Finding $a$ , the focus and the directrix

Step 2 — Solve for $a$

Finding $a$ , the focus and the directrix

Step 2 — Solve for $a$