Hyperbola

A hyperbola is a conic section. For every point on a hyperbola, the modulus (absolute value) of the difference of its distances from two fixed points is always the same. Incidentally, the word is a close cousin of “hyperbole”, which means exaggeration.

What a hyperbola looks like

That definition sounds a bit frightening, so first have a look at a picture of a hyperbola:

Notice that, unlike the other conic sections such as the ellipse or the parabola, a hyperbola is made of two curves. What does the definition mean? We have two foci, F₁ and F₂. For every point X on the hyperbola, the difference |XF₁|−|XF₂| must have the same absolute value.

In the picture we have two points, X₁ and X₂. For X₁ the difference is |X₁F₁|−|X₁F₂| = 1 − 3 = −2, and its absolute value is 2. We should get the same value for X₂. Let’s try: |X₂F₁|−|X₂F₂| = 3 − 5 = −2, with absolute value 2 (you can work out that the length of X₂F₂ is five using, for example, Pythagoras’ theorem).

If we apply this to all points of the hyperbola, we always get the result 2.

Description of a hyperbola

Have a look at an extended version of the previous picture:

• The points F₁ and F₂ are called the foci.
• The point S is called the centre of the hyperbola, and it lies at the midpoint of the line segment F₁F₂.
• The line F₁F₂ is called the transverse axis of the hyperbola. The perpendicular to this axis at S is called the conjugate axis.
• The points where the hyperbola meets the transverse axis are called the vertices of the hyperbola; in the picture they are A and B.
• Each of the segments AS and BS is called a semi-transverse axis. We write its length as a.
• We write the length of the semi-conjugate axis as b.
• The distance from a focus to the centre is called the linear eccentricity, written e. (Plain “eccentricity” means the ratio of this length to the semi-transverse axis.) The following relationship holds:

$$e=\sqrt{a^2+b^2}$$

To make it clearer where the length of the semi-conjugate axis b comes from, have a look at one more picture:

This is the same hyperbola, but now with the asymptotes added: the two crossed purple lines a₁, a₂ passing through the centre S. The semi-transverse axis a is unchanged, still the segment AS. Now we mark a point D on an asymptote so that the distance |SD| equals the linear eccentricity e. The length of the segment AD is then the length of the semi-conjugate axis. As the picture shows, this is a right-angled triangle, so we can use Pythagoras’ theorem, which gives

$$e=\sqrt{a^2+b^2}.$$

The circle used here has its centre at S and simply shows that the length |F₁S| (the linear eccentricity) is the same as |SD|.

The equation of a hyperbola

For a hyperbola we distinguish two different cases, depending on whether the transverse axis is parallel to the x-axis or the y-axis. Let the hyperbola have centre S with coordinates (m, n).

• The transverse axis is parallel to the x-axis:

Equation:

$$\frac{(x-m{)}^2}{a^2}-\frac{(y-n{)}^2}{b^2}=1$$

• The transverse axis is parallel to the y-axis:

Equation:

$$\frac{(y-n{)}^2}{a^2}-\frac{(x-m{)}^2}{b^2}=1$$