Parabola

A soccer ball kicked in an arc, following a parabolic path

When we kick a soccer ball (or shoot an arrow, fire a missile or throw a stone) it arcs up into the air and comes down again ...

... following the path of a parabola!

(Except for how the air affects it.)

Try kicking the ball:

images/parabola-ball.js?mode=ball

Parabola showing equal distances from a point on the curve to the focus and the directrix

Definition

A parabola is a curve where any point is at an equal distance from:

a fixed point (the focus), and
a fixed straight line (the directrix)

On Paper

Get a piece of paper, draw a straight line on it, then make a big dot for the focus (not on the line!).

Now play around with some measurements until you have another dot that's exactly the same distance from the focus and the straight line.

Keep going until you have lots of little dots, then join the little dots and you will have a parabola!

Just like in this interactive (try moving point P):

../sets/images/geom-locus.js?mode=parabola

Parabola with labels for directrix, vertex, focus, and axis of symmetry

Names

Here are the important names:

the directrix and focus (explained above)
the axis of symmetry (goes through the focus, at right angles to the directrix)
the vertex (where the parabola makes its sharpest turn) is halfway between the focus and directrix

Reflector

Parabolic reflector showing parallel incoming rays reflecting off the curve to meet at the focus

And a parabola has this amazing property:

Any ray parallel to the axis of symmetry gets reflected off the surface straight to the focus.

And that explains why that dot is called the focus ...

... because that's where all the rays get focused!

So the parabola can be used for:

A satellite dish, which is a real-world example of a parabolic reflector

satellite dishes,
radar dishes,
concentrating the sun's rays to make a hot spot,
the reflector on spotlights and torches,
and so on

A cone sliced by a plane parallel to its side, forming a parabola

We also get a parabola when we slice through a cone (the slice must be parallel to the side of the cone).

So the parabola is a conic section (a section of a cone).

Equations

Graph of the standard vertical parabola y = x^2 opening upwards

The simplest equation for a parabola is y = x²

Graph of the horizontal parabola y^2 = x opening right

Turned on its side it becomes y² = x

(or y = √x for just the top half)

Parabola y^2 = 4ax on coordinate axes showing focus at (a, 0) and directrix at x = -a

A little more generally:

y² = 4ax

where a is the distance from the origin to the focus (and also from the origin to directrix)

Where does y² = 4ax come from?

Looking at the diagram above with the vertex at the origin (0,0):

The focus is at (a, 0)
The directrix is the vertical line x = −a

For any point (x, y) on the parabola:

Distance to Focus (PF) =

√((x − a)² + y²)

Distance to Directrix (PM) =

x + a

Those distances are equal:

√((x − a)² + y²) = x + a

Square both sides:

(x − a)² + y² = (x + a)²

Expand the squares:

x² − 2ax + a² + y² = x² + 2ax + a²

Cancel same terms:

−2ax + y² = 2ax

Add 2ax to both sides:

y² = 4ax

Example: Find the focus for the equation y²=5x

Converting y² = 5x to y² = 4ax form, we get y² = 4 (5/4) x,

so a = 5/4, and the focus of y² = 5x is:

F = (a, 0) = (5/4, 0)

The equations of parabolas in different orientations are as follows:

Parabola opening right with equation y^2 = 4ax
y² = 4ax

Parabola opening left with equation y^2 = -4ax
y² = −4ax

Parabola opening upwards with equation x^2 = 4ay
x² = 4ay

Parabola opening downwards with equation x^2 = -4ay
x² = −4ay

Which way does it open?

When x is squared (like x² = 4ay), it opens up or down (like a bowl).
When y is squared (like y² = 4ax), it opens left or right (on its side)

Measurements for a Parabolic Dish

If you want to build a parabolic dish where the focus is 200 mm above the surface, what measurements do you need?

To make it easy to build, let's have it pointing upwards, and so we choose the x² = 4ay equation.

And we want "a" to be 200, so the equation becomes:

x² = 4ay = 4 × 200 × y = 800y

Rearranging so we can calculate heights:

y = x²/800

And here are some height measurements as you run along:

	Distance Along ("x")	Height ("y")
	0 mm	0.0 mm
	100 mm	12.5 mm
	200 mm	50.0 mm
	300 mm	112.5 mm
	400 mm	200.0 mm
	500 mm	312.5 mm
	600 mm	450.0 mm

Try to build one yourself, it could be fun! Just be careful, a reflective surface can concentrate a lot of heat at the focus.

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