The Hyperbola

If you change "sum of distances" to "difference of distances" in the ellipse definition, you get the hyperbola. The ellipse closes up into a single loop because adding two positive distances bounds the locus. The hyperbola opens out into two separate branches because the difference can grow without bound. Hyperbolas show up wherever something inversely proportional appears, the curves $y = k/x$ are hyperbolas in disguise, and they describe the paths of objects gravitationally slingshotting around the Sun fast enough to escape.

The standard equation

A hyperbola centred at the origin with transverse axis along the $x$-axis satisfies

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

The vertices are $(\pm a, 0)$; the transverse axis has length $2a$ and joins the two vertices through the centre. Notice the minus sign, the only algebraic difference between an ellipse and a hyperbola.

Asymptotes

For large $|x|$ the $1$ on the right becomes negligible and the equation tells you $y/x \to \pm b/a$. So the hyperbola approaches two straight lines:

$y = \pm \dfrac{b}{a} x$

These are the asymptotes. A handy construction trick: draw the central rectangle with corners at $(\pm a, \pm b)$. The asymptotes are the diagonals of this rectangle, and the hyperbola hugs them as it heads to infinity.

Foci and eccentricity

The foci sit on the transverse axis at distance $c$ from the centre, where

$c = \sqrt{a^2 + b^2}$ (note the plus sign)

The defining property: for any point on the hyperbola, the absolute difference of distances to the two foci equals $2a$. The eccentricity is $e = c/a > 1$, always greater than $1$, in contrast to the ellipse where $e < 1$.

• Navigation: LORAN (long-range navigation) systems located ships by measuring time differences between radio pulses from two stations; the locus of constant time-difference is a hyperbola with the stations as foci.
• Particle Physics: Rutherford's gold-foil experiment showed alpha particles deflected on hyperbolic paths in the field of the nucleus, the asymptotes give the scattering angle, and this is how the atomic nucleus was discovered.
• Economics: Indifference curves in microeconomics often have a hyperbolic shape; the asymptotes encode the limit consumption rates of two substitute goods.

(Choose "hyperbola" mode. Slide $a$ and $b$; the asymptotes tilt as you adjust their ratio. Watch the central rectangle widen and the foci move accordingly.)

Vertical transverse axis

If the equation is $y^2/a^2 - x^2/b^2 = 1$, the transverse axis is vertical: vertices at $(0, \pm a)$, asymptotes $y = \pm (a/b) x$. Which term is positive tells you which axis is transverse.

Try it

• Find the asymptotes and foci of $x^2/9 - y^2/16 = 1$.
• A hyperbola has vertices $(\pm 4, 0)$ and asymptotes $y = \pm \tfrac{3}{4} x$. Write its equation.
• What is the eccentricity of $x^2/4 - y^2/12 = 1$?

Pause: the rectangular hyperbola $xy = 1$ does not look like the standard form, but it is one, rotated by $45°$. After rotation, it becomes $(x')^2/2 - (y')^2/2 = 1$. The next section explores exactly this kind of axis rotation.

A trap to watch for

The sign in $c = \sqrt{a^2 + b^2}$ uses addition, opposite to the ellipse where it is subtraction. Beginners constantly write $c = \sqrt{a^2 - b^2}$ for hyperbolas and get nonsense (often imaginary numbers when $b > a$). Anchor it geometrically: an ellipse is contained inside its bounding rectangle, so its foci must be closer to the centre than the vertices, $c < a$, forcing subtraction. A hyperbola opens out beyond its vertices, so its foci are farther than the vertices, $c > a$, forcing addition.

A second trap: foci on the wrong axis. For $x^2/a^2 - y^2/b^2 = 1$ the transverse axis is horizontal and the foci are at $(\pm c, 0)$. For $y^2/a^2 - x^2/b^2 = 1$ they are at $(0, \pm c)$. Read the sign pattern, not the size of the denominators.

What you now know

You can write the standard hyperbola equation, find the asymptotes by inspection, locate the foci using $c = \sqrt{a^2 + b^2}$, compute the eccentricity (always $> 1$), and distinguish horizontal- and vertical-transverse-axis hyperbolas. The next section ties together all four conics and shows how to rotate the axes so that a tilted conic equation simplifies to standard form.

Quick check

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References

• Lang, S. (1971). Basic Mathematics. Springer. Chapter 12, §4, the hyperbola as the difference-of-distances locus.
• Apostol, T. M. (1969). Calculus, Volume 2. Wiley. Chapter 13: conic sections, including the unified focus-directrix-eccentricity view.
• Stewart, J. (2015). Calculus: Early Transcendentals (8th ed.). Cengage. Section 10.5: conic-section properties and applications.