Hyperbolas consist of two pieces that are shaped similarly to parabolas. One-piece opens up and the other down or one to the left and the other to the right. Also, similar to parabolas, each of the pieces has a vertex. The graphs of hyperbolas also have two lines, which are called asymptotes. The asymptotes are not officially part of the hyperbolas, but they are included to make sure we get the correct graph. The most important elements of a hyperbola are the foci, vertices, axes, focal length, semi-axes, and asymptotes.
Here, we will learn more details of these elements along with diagrams.
Definition of a hyperbola
A hyperbola is the set of all points, which have distances from two fixed points, called foci, which have a difference that is equal to a constant.
For example, if the points and are the foci and d is a constant, then the point is part of the hyperbola if as shown in the following picture:
We can also define hyperbolas as the conic sections that are formed by the intersection of two cones with an inclined plane that intersects the base of the cones. Hyperbolas consist of two separate curves, called branches.
The points at which the distance is the minimum between the two branches are called the vertices. The midpoint of the vertices of the hyperbola is the center. A hyperbola is asymptotic with respect to certain lines drawn through the center.
Fundamental elements of a hyperbola
The following are the fundamental elements of a hyperbola:
- Transverse axis
- Conjugate axis
- Semi-major axis
- Semi-minor axis
- Focal length
- Axes of symmetry
The foci are the fixed points used to define the hyperbola. The foci are often defined by and or also by and .
The coordinates of the foci are given by if the transverse axis is parallel to the x-axis and by if the transverse axis is parallel to the y axis. The point is the center and we find c using .
The transverse axis, also known as the real axis, is the segment that goes through the two foci. The transverse axis can be determined using the equation of the hyperbola. We know that the equation of the hyperbola contains a negative and a positive term.
If the positive sign is in the x term, it means that the transverse axis is in the x-axis and if the positive sign is in the y term, it means that the transverse axis is in the y axis.
The conjugate axis, also known as the imaginary axis, is the perpendicular bisector of the transverse axis. The conjugate axis divides the transverse axis into two equal parts.
The semi-major axis is the segment that extends from the center to a vertex of the hyperbola. Its length is denoted by a.
The semi-minor axis is the segment perpendicular to the semi-major axis. Its length is denoted by b.
The center has two lines of symmetry. The center is the point of intersection of the two lines of symmetry. If the hyperbola is centered at the origin, the center is (0,0) and if it is centered at another point, the center is .
The vertices are the points of intersection of the hyperbola with the transverse axis. The vertices are the endpoints of each branch of the hyperbola. Usually, we use and or and to represent the vertices.
The focal length is the length of the segment that extends from one focus () to the other focus (). Its length is equal to .
Axes of symmetry
The lines of symmetry are the axes that coincide with the transverse axis and the conjugate axis. The two branches of the hyperbola are symmetrical. Hyperbolas have two lines of symmetry, the horizontal axis, and the vertical axis. The point of intersection of these axes is the center.
The asymptotes are the lines that are very close to the branches of the hyperbola but never touch it. The asymptotes intersect at the center of the hyperbola.
Interested in learning more about the equations of a hyperbola? Take a look at these pages: