A hyperbola is a conical section formed by a plane that intersects both bases of the cone. Hyperbolas consist of two asymptotic branches to two intersecting fixed lines. The point of intersection of the lines is the center of the hyperbola. In addition to asymptotes, hyperbolas also have other parts that serve to define them. Some of the important parts of hyperbolas are the foci, the vertices, the axes, the semi-axes, and the focal length.
Here, we learn about the parts of hyperbolas in more detail. We will use diagrams to illustrate the concepts.
What are hyperbolas?
Hyperbolas are curves formed by the set of all points, which are characterized by having distances from two fixed points that produce a difference that is equal to a constant. The two fixed points are called the foci of the hyperbolas.
For example, we can define the points and as the foci, and d represents a constant. For any point with coordinates to be part of the hyperbola, we must have . This is illustrated in the following diagram:
Additionally, hyperbolas are defined as the conic sections formed when a plane intersects a pair of cones. For a hyperbola to be formed, the plane must be inclined with respect to the base of the cone and must cross at both bases.
Hyperbolas are made up of two curves, called branches. Each of these branches is shaped like a parabola and each has a vertex.
Important parts of a hyperbola
The following are the most important parts of a hyperbola:
- Focal length
- Transverse axis
- Conjugate axis
- Axes of symmetry
- Semi-major axis
- Semi-minor axis
The foci are two fixed points that define the hyperbola. Each branch of the hyperbola contains a focus. We use the letter to represent the foci. We can use the notation and to distinguish them or also the notation and .
We use the value c to find the coordinates of the foci, which are given by if the hyperbola is oriented horizontally or if the hyperbola is oriented vertically, where the point is the center.
Furthermore, we can find the value of c using .
The focal length is the length of the distance between the two foci. This length is equal to 2c.
The transverse axis is also known as the real axis. This axis is the segment that connects the two foci. The position of the transverse axis is found using the equation of the hyperbola. Since all hyperbolas are made up of a negative and a positive term, we can distinguish two cases.
When the x term is the positive part, the transverse axis is on the x-axis or is parallel to the x-axis. When the y term is the positive part, the transverse axis is on the y-axis or is parallel to the y-axis.
The conjugate axis is also known as the imaginary axis. This axis is perpendicular to the transverse axis and divides it into two equal parts.
Axes of symmetry
The hyperbola has two lines of symmetry, a horizontal axis, and a vertical axis. These axes coincide with the transverse axis and the conjugate axis.
The center is the point where the two lines of symmetry of the hyperbola intersect. The center is also the point of intersection of the two asymptotes. When a hyperbola is centered at the origin, the center is (0,0) and when it is centered outside the origin, the center is .
The vertices are the endpoints of the transverse axis. These points are located at the intersection of the hyperbola and the transverse axis. We use to represent the vertices and we can distinguish them using and or and .
The semi-major axis is a line segment that connects the center to a vertex of the hyperbola. Usually, we denote its length using a.
The semi-minor axis is a line segment that is perpendicular to the semi-major axis. Usually, we denote its length using b.
The asymptotes represent the behavior of the hyperbolas. The asymptotes are the lines that are very close to the branches of the hyperbola but never touch it.
Interested in learning more about hyperbolas? Take a look at these pages: