The Hyperbola

Stretch a rubber band around two pins and trace the curve that keeps the sum of the two distances fixed and you get an ellipse. The hyperbola is its rebellious twin: instead of the sum of the distances to two special points, it keeps the difference fixed. Same two points — the foci — but a mirror-image rule, and a wildly different shape.

A hyperbola is every point (x, y) for which the distances to two fixed foci F_1 and F_2 differ by a constant amount, written 2a:

\left| \, PF_1 - PF_2 \, \right| = 2a.

Because a difference can be positive or negative, this rule carves out two separate curves — the two branches, one hugging each focus. That is the hyperbola's signature: it comes in a matched pair that open away from each other and race off toward infinity.

The standard equation

Put the two foci on the x-axis, symmetric about the origin, and the "constant difference" rule tidies up into a single clean equation:

\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1.

The minus sign is everything — swap it for a plus and you would be back at an ellipse. Reading this equation gives you the whole figure at a glance:

The letter under the positive term is a, and it always points along the axis the curve opens toward. Here x^2 is positive, so the branches open left and right.

Meet the asymptotes

Follow a branch far enough from the centre and it flattens against a pair of straight lines it never quite touches — the asymptotes. For \dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1 they are the two lines

y = \pm \frac{b}{a}\, x.

They are the single most useful sketching aid you own: draw the box a wide and b tall around the centre, run the diagonals out to infinity, and the branches slot neatly inside, kissing the diagonals as they go. Step through the figure below to watch the branches, vertices, asymptotes and foci appear one layer at a time.

Worked examples

1 · Read off everything from \dfrac{x^2}{9} - \dfrac{y^2}{16} = 1. Match it to \dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1. The number under x^2 is a^2 = 9, so a = 3; the number under y^2 is b^2 = 16, so b = 4. Now use the plus relation:

c^2 = a^2 + b^2 = 9 + 16 = 25 \;\Rightarrow\; c = 5.

So the vertices are (\pm 3, 0), the foci are (\pm 5, 0), the asymptotes are y = \pm \frac{4}{3} x, and the eccentricity is

e = \frac{c}{a} = \frac{5}{3} \approx 1.67 \; (> 1).

2 · Asymptotes and foci of \dfrac{x^2}{4} - \dfrac{y^2}{9} = 1. Here a^2 = 4 and b^2 = 9, so a = 2 and b = 3. The asymptotes are y = \pm \frac{b}{a} x = \pm \frac{3}{2} x. For the foci,

c^2 = a^2 + b^2 = 4 + 9 = 13 \;\Rightarrow\; c = \sqrt{13} \approx 3.61,

so the foci are (\pm \sqrt{13},\, 0). The c need not be a whole number — only c^2 has to be.

Other flavours

Swap the roles of x and y and the hyperbola stands up on end:

\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1

now has the y^2 term positive, so it opens up and down, with vertices at (0, \pm a) and foci at (0, \pm c). The rule for which way it opens is always the same: whichever variable sits over the positive term is the axis it opens along.

There is also the friendly rectangular hyperbola xy = k — the graph of inverse proportion, y = k/x. Its two branches live in opposite quadrants and its asymptotes are simply the x- and y-axes. It is the same curve, just rotated 45^\circ.

Here is the beautiful part: the "constant difference of distances" rule is not just geometry — it is how navigation systems find you. Two radio stations broadcast synchronised pulses. A receiver can't measure how far it is from each station, but it can measure the difference in arrival times — and time difference means distance difference. A fixed distance difference is exactly the locus of a hyperbola with the two stations as foci, so your receiver knows it lies somewhere on one specific hyperbola.

Add a second pair of stations and you get a second hyperbola; where the two curves cross is your position. This was the whole idea behind LORAN (Long Range Navigation), which guided ships and aircraft for decades, and the same time-difference-of-arrival trick still helps modern GPS and mobile-phone location work. Every time a system turns "how much later did this signal arrive" into "where am I," a hyperbola is quietly doing the maths.

You have probably seen the shape without noticing, too: the curved edge of the shadow a lampshade throws on a wall, and the graceful waist of a power-station cooling tower, are both hyperbolas.