Hyperbola

A hyperbola is a cone. For each point of the hyperbola, the absolute value of the difference in distances from two fixed points is always the same. By the way, in English, hyperbola is another name for hyperbole.

What a hyperbola looks like

The previous definition sounds a bit scary, so let's look at a picture of a hyperbola first:

Notice that unlike other conics like an ellipse or a parabola, a hyperbola is composed of two curves. What does the previous definition mean? We are given two foci, F₁ and F₂. For each point X on the hyperbola, it must hold that the difference |XF₁|−|XF₂| is the same in absolute value.

In the figure, we have two points X₁ and X₂. For X₁, the difference comes out to be: |X₁F₁|−|X₁F₂| = 1 − 3 = −2, in absolute value, then we get the result 2. We should get the same value for X₂. Let's try: |X₂F₁|−|X₂F₂| = 3 − 5 = −2, in absolute value 2 (that the length of the side X₂F₂ is equal to five you can calculate, for example, using the Pythagorean theorem).

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

Description of the hyperbola

Look at the expanded image of the previous hyperbola:

• The points F₁ and F₂ are called foci.
• The point S is called the center of the hyperbola and is located at the center of the line segment F₁F₂.
• The line F₁F₂ is called the major axis of the hyperbola. The perpendicular to this axis at the point S is called the minor axis of the hyperbola.
• The intersections of the hyperbola with the major axis are called the vertices of the hyperbola, in the figure they are the points A and B.
• The segments AS and BS are called the major axis of the hyperbola. We denote their length by a.
• We denote the length of the minor semi-axis of the hyperbola by b.
• The distance of the focus from the center is called the eccentricity, we denote by e. The relation holds:

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

To see better where the length of the minor semi-axis b came from , look at one more figure:

This is the same hyperbola, where, first of all, the asymptotes have been added, which are the two crossed purple lines a₁, a₂ that pass through the center S. The major semi-axis a remains unchanged, it is still a line AS. But now we plot a point D on the asymptote so that the distance |SD| is equal to the eccentricity e. The length of the line segment AD then represents the length of the minor semi-axis of the hyperbola. As you can see from the figure, this is a right triangle, so we can use the Pythagorean theorem and therefore the relation

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

The circle used has a center at S and just shows that the length of |F₁S| (eccentricity) is the same as the length of |SD|.

The equation of the hyperbola

For the hyperbola, we distinguish two different cases. It depends on whether the major axis of the hyperbola is parallel to the axis x or to y. Consider a hyperbola centered at S with coordinates [m, n].

• Themajor axis is parallel to the axis x:

Equation:

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

• Theprincipal axis is parallel to the axis y:

Equation:

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