A hyperbola is the locus of a point in a plane which moves in the plane in such a way that the ratio of its distance from a fixed point (called focus) in the same plane to its distance from a fixed line (called directrix) is always constant which is always greater than unity.
The constant ratio is generally denoted by ‘e’ and is known as the eccentricity of the hyperbola.
Every hyperbola has a second focus and second directrix.
The difference of the focal distances of any point on a hyperbola is constant and is equal to the length of transverse axis of the hyperbola.
On account of this property, a second definition of the hyperbola is:
A hyperbola is the locus of a point which moves in such a way that the difference of its distance from two fixed points (foci) is always constant.
Conjugate hyperbola: The hyperbola whose transverse and conjugate axes are respectively the conjugate and transverse axes of a given hyperbola is called the conjugate hyperbola of the given hyperbola.
Director circle: is the locus of points from which perpendicular tangents are drawn to the given hyperbola.
Asymptotes: An asymptote to a curve is a straight line, at a finite distance from the origin, to which the tangent to a curve tends as the point of contact goes to infinity. In other words, asymptote to a curve touches the curve at infinity.
Rectangular hyperbola: A hyperbola whose asymptotes are right angles to each other is called a rectangular hyperbola.