The Parabola

Slice a cone with a flat plane and you get a family of curves called the conic sections. Tilt the plane just so — parallel to the slope of the cone — and the cut you expose is a parabola: the graceful open curve of a thrown ball, a fountain's arc, or the cable of a suspension bridge. You have met one already as the graph of a quadratic, y = ax^2. This page reveals what that curve really is — and why it hides inside every satellite dish and car headlight on Earth.

A circle is every point a fixed distance from one point. A parabola swaps that single centre for a point and a line, and balances between them:

A parabola is the set of all points that are the same distance from a fixed point (the focus) as from a fixed line (the directrix). \text{distance to the focus} \;=\; \text{distance to the directrix}.

Every point on the curve keeps this promise exactly; no point off the curve does. The focus sits inside the bend of the curve, the directrix runs outside it, and the curve threads perfectly between them.

The two equal distances, in a picture

Step through the figure below. A point P on the parabola sends one segment straight to the focus and one straight down to the directrix. Drag your eye along the curve in your mind: those two lengths grow and shrink together, staying equal at every single point. That equality is the parabola.

The standard equations

Put the vertex at the origin and the equal-distance rule turns into tidy algebra. Let p be the distance from the vertex to the focus (and, by symmetry, from the vertex to the directrix). There are two orientations.

The number 4p is the whole personality of the parabola: a big p means a far focus and a wide, lazy curve; a small p means a near focus and a tight, narrow one.

The familiar y = ax^2, decoded

The school parabola y = ax^2 is exactly x^2 = 4py in disguise. Rearrange: x^2 = \tfrac{1}{a}\,y, so matching 4p = \tfrac{1}{a} gives

a = \frac{1}{4p} \qquad\Longleftrightarrow\qquad p = \frac{1}{4a}.

So the plain graph y = x^2 (with a = 1) has its focus a mere p = \tfrac14 above the vertex, at \left(0, \tfrac14\right), and directrix y = -\tfrac14. The steepness you already know and the focal distance are two sides of the same coin.

Worked examples

1 · Read off the focus and directrix. Take y^2 = 8x. This is the horizontal form y^2 = 4px, so match the coefficients:

4p = 8 \;\Longrightarrow\; p = 2.

Therefore the focus is at (p, 0) = (2, 0) and the directrix is the line x = -p, i.e. x = -2. The curve opens to the right (because p > 0), wrapping around its focus.

2 · Build the equation from a focus and directrix. Find the parabola with focus (0, 3) and directrix y = -3. The focus is on the y-axis and the directrix is horizontal, so this is the vertical form x^2 = 4py. Here the focus is (0, p), so p = 3, and

x^2 = 4py = 4(3)y = 12y.

Check it against the definition: the point (6, 3) lies on x^2 = 12y (since 36 = 12 \cdot 3). Its distance to the focus (0,3) is 6; its distance to the directrix y = -3 is 3 - (-3) = 6. Equal — as promised.

Spin a parabola around its axis and you get a paraboloid — a dish. It has one magical habit, the reflective property: every ray coming in parallel to the axis bounces off the surface and passes through a single point — the focus. (And run it backwards: a bulb at the focus throws out a perfectly parallel beam.)

That one fact powers a surprising amount of the world. A satellite dish gathers faint parallel signals from space and concentrates them onto a receiver placed exactly at the focus. A reflecting telescope does the same with starlight. Run the idea in reverse and a car headlight or torch puts its bulb at the focus to hurl a straight, bright beam down the road. Even a solar cooker uses it to pile sunlight onto one blazing hot spot. The word "focus" is Latin for hearth — the fireplace — and now you know why.

The tempting mistake is to read the steepness a as the distance to the focus. It is not — in fact it is upside down. The focal distance is p = \tfrac{1}{4a}, so a and p pull in opposite directions:

And don't confuse the two constants across the forms: in y^2 = 4px the coefficient is 4p (four times the focal distance), whereas in y = ax^2 the coefficient a is its reciprocal over four. Same curve, two different bookkeepers.

Once you have the parabola, its cousins are a short step away: the closed ellipse and the two-branched hyperbola are the other slices of the same cone.