Introduction to Hyperbolas

A hyperbola can be defined in a number of ways. A hyperbola is: 

1. The intersection of a right circular double cone with a plane perpendicular to the base of the cone
2. The set of all points such that the difference between the distances to two focal points is constant
3. The set of all points such that the ratio of the distance to a single focal point divided by the distance to a line (the directrix) is greater than one

Let's see how that second definition gives us what is called the standard form of a hyperbola equation.

Diagram of a hyperbola

The hyperbola, shown in blue, has a center at the origin, two focal points at $(-c,0)$ and $(c,0)$, and two vertices located at $+a$ and $-a$ on the $x$-axis.

We begin with two focal points, $F_1$ and $F_2$, located on the $x$-axis, so that they have coordinates $(c,0)$ and $(-c,0)$ (other arrangements are possible). We want the set of all points that have the same difference between the distances to these points. The center of this hyperbola is the origin $(0,0)$.

Imagine that we take a point on the red hyperbola curve, called $P$, and we let that point be the $+a$ value on the $x$-axis. Then the difference of distances between $P$ and the two focal points is:

$\displaystyle{ \begin{aligned} (P \rightarrow F_2) - (P \rightarrow F_1) &= (c+a) - (c - a)\\ &= 2a \end{aligned} }$

where $a$ is the distance from the center (origin) to the vertices of the hyperbola. With this value for the difference of distances, we can choose any point $(x,y)$ on the hyperbola and construct an equation by use of the distance formula:

$\displaystyle{ \begin{aligned} \sqrt{ (x-c)^2 + (y-0)^2} - \sqrt{ (x - (-c))^2 + (y-0)^2 } &= 2a \\ \sqrt{ (x-c)^2 + y^2} - \sqrt{(x+c)^2 + y^2} &= 2a \end{aligned} }$

From here there is some straightforward, but messy, algebra. We need to square both sides of this equation multiple times if we want the variables to escape their square roots. When the dust settles, we have:

$x^2(c^2 - a^2) - a^2y^2 = a^2(c^2 - a^2)$

At this point we introduce one more parameter, defined as $b^2 = c^2 - a^2$, which reduces the hyperbola even further:

$b^2x^2 - a^2y^2 = a^2b^2$

Lastly we divide both sides of the equation by $a^2b^2$:

$\displaystyle{ \begin{aligned} b^2x^2 - a^2y^2 &= a^2b^2 \\ \frac{b^2x^2}{a^2 b^2} - \frac{a^2y^2}{a^2 b^2} &= \frac{a^2 b^2}{a^2 b^2} \\ \frac{x^2}{a^2} - \frac{y^2}{b^2} &= 1 \end{aligned} }$

Thus, the standard form of the equation for a hyperbola with focal points on the $x$ axis is:

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

If the focal points are on the $y$-axis, the variables simply change places:

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

Note that the hyperbola standard form is very similar to the standard form of the ellipse:

$\displaystyle{\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1}$

The similarity is not coincidental. The ellipse can be defined as all points that have a constant sum of distances to two focal points, and the hyperbola is defined as all points that have constant difference of distances to two focal points.

There is another common form of hyperbola equation that, at first glance, looks very different: $\displaystyle{y = \frac{1}{x}}$ or $xy = 1$.

Reciprocal hyperbola

This hyperbola is defined by the equation $xy = 1$.

From the graph, it can be seen that the hyperbola formed by the equation $xy = 1$ is the same shape as the standard form hyperbola, but rotated by $45^\circ$. To prove that it is the same as the standard hyperbola, you can check for yourself that it has two focal points and that all points have the same difference of distances. Another way to prove it algebraically is to construct a rotated $x$-$y$ coordinate frame.