A hyperbola is a type of conic section formed by the intersection of a plane and a double cone. It consists of two separate curves called branches, which open away from each other. The standard equation of a hyperbola can be expressed as (x^2/a^2) - (y^2/b^2) = 1 for horizontal hyperbolas, and (y^2/b^2) - (x^2/a^2) = 1 for vertical hyperbolas, where a and b are constants that determine the shape and size of the hyperbola. Hyperbolas have unique properties, including asymptotes, which are lines that the branches approach but never touch. They are often used in various fields such as mathematics, physics, and engineering to model certain types of behavior, such as the paths of objects in orbital mechanics. Hyperbolas also appear in