In mathematics, a hyperbola is a type of conic section, just like an ellipse. It is defined as the set of points in a plane where the difference of the distances to two fixed points, called the foci, is constant. A hyperbola has two distinct branches that are symmetric about its center.
The equation of a hyperbola in standard form, centered at the origin, is given by:
(x² / a²) - (y² / b²) = 1
In this equation, 'a' represents the distance from the center to each vertex along the x-axis, and 'b' represents the distance from the center to each vertex along the y-axis. The foci of the hyperbola are located at a distance of c units from the center, where c is given by the equation c = √(a² + b²).
The major axis of a hyperbola corresponds to the x-axis (for a horizontal hyperbola) or the y-axis (for a vertical hyperbola). The vertices of the hyperbola are located on the major axis.
In terms of differences from an ellipse, there are a few key distinctions:
Shape: A hyperbola has two separate branches, whereas an ellipse is a closed curve with a single continuous boundary.
Eccentricity: The eccentricity of a hyperbola is always greater than 1, indicating a more elongated shape. In contrast, the eccentricity of an ellipse is between 0 and 1, with 0 representing a circle.
Symmetry: While both hyperbolas and ellipses possess symmetry about their respective centers, the axes of symmetry differ. For a hyperbola, the symmetry axes coincide with the transverse and conjugate axes, whereas for an ellipse, they align with the major and minor axes.
In physics, hyperbolas and ellipses can also describe the paths of objects in certain physical systems. For example, in the case of planetary motion, Kepler's laws state that planets orbit the Sun in elliptical paths. On the other hand, in certain scenarios involving the interaction of particles, such as the gravitational or electrostatic forces, hyperbolic trajectories can arise for objects with sufficient energy or velocity.