Have you ever seen the shape of cooling towers at power plants? Or perhaps noticed the shadow of a flashlight on the wall forming an open curve? Well, shapes like these are examples of hyperbolas in real life!
A hyperbola is a curve formed when a plane cuts through a double cone at a certain angle. Unlike an ellipse which forms a closed curve, a hyperbola is actually formed from two separate curves that face each other.
Mathematically, a hyperbola is defined as the locus of points where the absolute difference of distances to two fixed points is always constant. These two fixed points are called the foci of the hyperbola. For every point P on the hyperbola, the difference ∣PF1−PF2∣=2a (constant), whereas in an the sum of distances is constant: .
Hyperbola with two foci showing constant difference in distances
Look at the visualization above! A hyperbola has two separate branches. For every point P on the hyperbola, the difference in distance from P to both foci F1 and F2 is always constant.
Before diving into formulas, let's get acquainted with the important parts of a hyperbola. Each component has its own role in determining the shape and properties of the hyperbola.
Parts of a Hyperbola
Important components of a hyperbola with horizontal major axis.
The components of a hyperbola you need to know:
Center of hyperbola is the midpoint between the two foci, usually denoted by O.
Foci (F1 and F2) are two fixed points that serve as the reference for the definition of a hyperbola. The distance between the two foci is called the focal distance.
Vertices (A1 and A2) are the closest points between the two branches of the hyperbola. The line connecting the two vertices is called the major axis.
Major axis is the line that passes through the center and both foci of the hyperbola.
Asymptotes are lines that are approached by the branches of the hyperbola as they extend to infinity. The hyperbola never touches its asymptotes, but the farther from the center, the closer the curve approaches the asymptote lines.
The main difference between a hyperbola and an ellipse: a hyperbola has asymptotes and consists of two separate branches, while an ellipse is a closed curve without asymptotes.
There are several mathematical relationships that always hold for every hyperbola:
c2=a2+b2
where:
a is the distance from center to vertex (semi-major axis)
b is the constant that determines the shape of the hyperbola (semi-conjugate axis)
c is the distance from center to focus
This relationship is different from an ellipse which uses c2=a2−b2.
Eccentricity of a hyperbola is defined as:
e=ac
The eccentricity value of a hyperbola is always e>1. This is because c>a (from the relationship c2=a2+b2, so c=a2+b2>a). The larger the value of e, the more "open" or wide the hyperbola becomes. For comparison: a circle has e=0, an ellipse has 0<e<1, a parabola has e=1, and a hyperbola has e>1.
Asymptote equations for a hyperbola with center at (0,0):
y=±abx(horizontal major axis)
y=±bax(vertical major axis)
Asymptotes are "guides" for the hyperbola branches. The farther from the center, the closer the hyperbola curve approaches the asymptote lines, but never touches them.
