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 $latex F_{1}$ and $latex F_{2}$ are the foci and d is a constant, then the point $latex (x, ~y)$ is part of the hyperbola if $latex d =|d_{1}-d_{2}|$ 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:
- Foci
- Transverse axis
- Conjugate axis
- Semi-major axis
- Semi-minor axis
- Center
- Vertices
- Focal length
- Axes of symmetry
- Asymptotes
Foci
The foci are the fixed points used to define the hyperbola. The foci are often defined by $latex F_{1}$ and $latex F_{2}$ or also by $latex F$ and $latex F’$.
The coordinates of the foci are given by $latex F=(h \pm c, k)$ if the transverse axis is parallel to the x-axis and by $latex F = (h, k \pm c)$ if the transverse axis is parallel to the y axis. The point $latex (h, k)$ is the center and we find c using $latex {{c}^2}={{a}^2}+{{b}^2}$.
Transverse axis
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.
Conjugate 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.
Semi-major axis
The semi-major axis is the segment that extends from the center to a vertex of the hyperbola. Its length is denoted by a.
Semi-minor axis
The semi-minor axis is the segment perpendicular to the semi-major axis. Its length is denoted by b.
Center
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 $latex (h, k)$.
Vertices
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 $latex V_{1}$ and $latex V_{2}$ or $latex V$ and $latex V’$ to represent the vertices.
Focal length
The focal length is the length of the segment that extends from one focus ($latex F_{1}$) to the other focus ($latex F_{2}$). Its length is equal to $latex 2c$.
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.
Asymptotes
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.
See also
Interested in learning more about the equations of a hyperbola? Take a look at these pages:
