Hyperbola: Asymptotes, Foci, and the Constant Difference

A hyperbola is the set of all points where the absolute difference of distances to two foci is constant. Unlike an ellipse (constant sum), a hyperbola has two separate branches that curve away from each other, approaching but never touching their asymptotes.

Standard form (horizontal transverse axis)

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

• Vertices at $(\pm a, 0)$
• Asymptotes: $y = \pm (b/a)x$
• Foci at $(\pm c, 0)$ where $c = \sqrt{a^{2} + b^{2}}$ (note: $+$ not $-$, unlike ellipse)
• Constant difference: $|PF_{1} - PF_{2}| = 2a$

For $x^{2}/16 - y^{2}/9 = 1$: $a = 4$, $b = 3$, $c = \sqrt{16 + 9} = 5$.

• Vertices: $(\pm 4, 0)$
• Asymptotes: $y = \pm (3/4)x$
• Foci: $(\pm 5, 0)$
• Eccentricity: $e = c/a = 5/4 = 1.25$ (always $> 1$ for hyperbolas)

Asymptotes: the guiding lines

The asymptotes form an "X" through the center. As $|x| \to \infty$, the hyperbola curves approach these lines arbitrarily closely but never cross them. They define the "opening angle" of the hyperbola.

To find the asymptotes, set the equation equal to 0 instead of 1:

$\dfrac{x^{2}}{a^{2}} - \dfrac{y^{2}}{b^{2}} = 0 \implies y = \pm\dfrac{b}{a}x$

Real-world hyperbolas

• LORAN navigation: Ships determine position by measuring the time difference of radio signals from two stations. Each constant time difference defines a hyperbola; two pairs of stations give two hyperbolas whose intersection is the ship's position.
• Sonic booms: The shock wave from a supersonic aircraft forms a cone. The intersection of that cone with the ground is a hyperbola.
• Orbital trajectories: An object with more than escape velocity follows a hyperbolic orbit around a gravitational body (it doesn't come back).

Try it in the visualization

Adjust $a$ and $b$ to change the hyperbola's shape. The asymptotes update as guide lines. Toggle "constant difference" to see the two distance lines from a point on the curve to the foci — their absolute difference is always $2a$. Switch orientation between horizontal and vertical transverse axis.