Hyperbola Formulas

Hyperbolic Eccentricity Formula: A hyperbola’s eccentricity is always greater than 1, i.e. e > 1. The ratio of the distance of the point on the hyperbole from the focus to its distance from the directrix is the eccentricity of a hyperbola.

Eccentricity = Distance from Focus/Distance from Directrix 

or

e = c/a

We get the following value of eccentricity by substituting the value of c.

e =√(1+b²/a²)

Equation of Major axis: The Major Axis is the line that runs through the center, hyperbola’s focus, and vertices. 2a is considered the length of the major axis. The formula is as follows: 

y = y₀

Equation of Minor axis: The Minor Axis is a line that runs orthogonal to the major axis and travels through the middle of the hyperbola. 2b is the length of the minor axis. The following is the equation:

x = x₀

Asymptotes: The Asymptotes are two bisecting lines that pass through the center of the hyperbola but do not touch the curve. The following is the equation:

y = y₀ + (b/a)x – (b/a)x₀

y = y₀ – (b/a)x + (b/a)x₀

Directrix of a hyperbola: A hyperbola’s directrix is a straight line that is utilized to generate a curve. It is also known as the line away from which the hyperbola curves. The symmetry axis is perpendicular to this line. The directrix equation is:

x = ±a² /√(a² + b² )

Vertex: The vertex is the point on a stretched branch that is closest to the center. These are the vertex points.

[a, y₀ ] ; [-a, y₀ ]

Focus (foci): Focus ( foci) are the fixed locations on a hyperbola where the difference between the distances is always constant.

(x₀ + √(a² + b²), y₀)

(x₀ – √(a² + b²), y₀)

Conjugate Hyperbola: Two hyperbolas whose transverse and conjugate axes are the conjugate and transverse axes of the other are referred to as conjugate hyperbolas of each other.

(x² / a²) – (y² /b²) = 1 and  (−x² / a²) + (y² / b²) = 1 

are conjugate hyperbolas together each other. Therefore:

(y² / b²) – (x² / a²) = 1

a² = b² (e² − 1)

e = √( 1+ a²/b²)

Hyperbola Formula: The set of all points in a plane is called a hyperbola. The distance between these two fixed points in the plane will remain constant. The distance to the distant location minus the distance to the nearest point is the difference. The foci will be the two fixed points, and the center of the hyperbola will be the mid-point of the line segment connecting the foci. Hyperbola is a fascinating topic in geometrical mathematics.

This article explores the hyperbola formulas, along with their equations, and solved examples on it.