Hyperbola
Hyperbola is an important conic formed by the intersection of the double cone by a plane surface, but not necessarily at the center. A hyperbola is symmetric along the conjugate axis, and share many similarities with the ellipse. Concepts like foci, directrix, latus rectum, eccentricity, apply to a hyperbola. A few common examples of hyperbola include the path followed by the tip of the shadow of a sundial, the scattering trajectory of subatomic particles.
Here we shall aim at understanding the definition, formula of a hyperbola, derivation of the formula, standard forms of hyperbola, also check through the solved examples, frequently asked questions.
1.  Hyperbola  Definition 
2.  Derivation  Hyperbola Equation 
3.  Standard Equations of a Hyperbola 
4.  Properties of a Hyperbola 
5.  Solved Examples 
6.  Practice Questions 
7.  FAQs on Hyperbola 
Hyperbola  Definition
A hyperbola is a set of points whose difference of distances from two foci is a constant value. This difference is taken from the distance from the farther focus and then the distance from the nearer focus. For a point P(x, y). on the hyperbola and for two foci F, F', the locus of the hyperbola is PF  PF' = 2a. The below equation represents the standard form of the equation of a hyperbola. Here the xaxis is the transverse axis of the hyperbola, and the yaxis is the conjugate axis of the hyperbola.
\(\dfrac{x^2}{a^2}  \dfrac{y^2}{b^2} = 1\)
Let us check through a few important terms relating to a hyperbola.
 Focus: The hyperbola has two foci and their coordinates are F(c, o), and F'(c, 0).
 Center: The midpoint of the line joining the two foci is called the center of the hyperbola.
 Major Axis: The length of the major axis of the hyperbola is 2a units.
 Minor Axis: The length of the minor axis of the hyperbola is 2b units.
 Vertices: The points where the hyperbola intersects the axis are called the vertices. The vertices of the hyperbola are (a, 0), (a, 0).
 Latus Rectum: The latus rectum is a line drawn perpendicular to the transverse axis of the hyperbola and is passing through the foci of the hyperbola. The length of the latus rectum of the hyperbola is 2b^{2}/a.
 Transverse Axis: The line passing through the two foci and the center of the hyperbola is called the transverse axis of the hyperbola.
 Conjugate Axis: The line passing through the center of the hyperbola and perpendicular to the transverse axis is called the conjugate axis of the hyperbola.
 Eccentricity: (e > 1) The eccentricity is the ratio of the distance of the focus from the center of the ellipse, and the distance of the vertex from the center of the ellipse. The distance of the focus is 'c' units, and the distance of the vertex is 'a' units, and hence the eccentricity is e = c/a.
Derivation  Hyperbola Equation
As per the definition of the hyperbola, let us consider a point P on the hyperbola, and the difference of its distance from the two foci F, F' is 2a.
PF'  PF = 2a
Let the coordinates of P be (x, y) and the foci be F(c, o) and F'(c, 0)
\(\sqrt{(x + c)^2 + y^2}\)  \(\sqrt{(x  c)^2 + y^2}\) = 2a
\(\sqrt{(x + c)^2 + y^2}\) = 2a + \(\sqrt{(x  c)^2 + y^2}\)
Now we need to square on both sides to solve further.
(x + c)^{2} + y^{2} = 4a^{2 }+ (x  c)^{2} + y^{2} + 4a\(\sqrt{(x  c)^2 + y^2}\)
x^{2} + c^{2} + 2cx + y^{2} = 4a^{2 }+ x^{2} + c^{2}  2cx + y^{2} + 4a\(\sqrt{(x  c)^2 + y^2}\)
4cx  4a^{2} = 4a\(\sqrt{(x  c)^2 + y^2}\)
cx  a^{2} = a\(\sqrt{(x  c)^2 + y^2}\)
Squaring on both sides and simplifying, we have.
\(\dfrac{x^2}{a^2}  \dfrac{y^2}{c^2  a^2} =1\)
Also we have c^{2} = a^{2} + b^{2} we can substitute this in the above equation.
\(\dfrac{x^2}{a^2}  \dfrac{y^2}{b^2} =1\)
This derives the standard equation of the Hyperbola.
Standard Equations of a Hyperbola
There are two standard equations of the Hyperbola. These equations are based on the transverse axis and the conjugate axis of each of the hyperbola. The standard equation of the hyperbola is \(\dfrac{x^2}{a^2}  \dfrac{y^2}{b^2} = 1\) has the transverse axis as the xaxis and the conjugate axis is the yaxis. Further, another standard equation of the hyperbola is \(\dfrac{x^2}{b^2} + \dfrac{y^2}{a^2} = 1\) and it has the transverse axis as the yaxis and its conjugate axis is the xaxis. The below image shows the two standard forms of equations of the hyperbola.
Properties of a Hyperbola
The following important concepts help in understanding the properties of a hyperbola.
Asymptotes: The pair of straight lines drawn parallel to the hyperbola and assumed to touch the hyperbola at infinity. The equations of the asymptotes of the hyperbola are y = bx/a, and y = bx/a respectively. And the equation of the pair of asymptotes of the hyperbola are \(\dfrac{x^2}{a^2}  \dfrac{y^2}{b^2} = 0\).
Rectangular Hyperbola: The hyperbola having the transverse axis and the conjugate axis of the same length is called the rectangular hyperbola. Here we have 2a = 2b, or a = b. Hence the equation of the rectangular hyperbola is equal to x^{2}  y^{2} = a^{2}
Parametric Coordinates: The points on the hyperbola can be represented with the parametric coordinates (x, y) = (asecθ, btanθ). These parametric coordinates representing the points on the hyperbola satisfy the equation of the hyperbola.
Auxilary Circle: A circle drawn with the endpoints of the transverse axis of the hyperbola as its diameter is called the auxiliary circle. The equation of the auxiliary circle of the hyperbola is x^{2} + y^{2} = a^{2}
Direction Circle: The locus of the point of intersection of perpendicular tangents to the hyperbola is called the director circle. The equation of the director circle of the hyperbola is x^{2} + y^{2} = a^{2}  b^{2}
Solved Examples on Hyperbola

Example 1: The equation of the hyperbola is given as (x−5)^{2}/4^{2}−(y−2)^{2}/ 6^{2} = 1. Use the hyperbola formulas to find the length of the Major Axis and Minor Axis.
Solution:
Using the hyperbola formula for the length of the major and minor axis
Length of major axis = 2a, and length of minor axis = 2b
Length of major axis = 2×4 = 8, and Length of minor axis = 2×6 = 12
Answer: The length of the major axis is 8 units, and the length of the minor axis is 12 units.

Example 2: The equation of the hyperbola is given as (x−5)^{2}/4^{2}−(y−2)^{2}/ 6^{2} = 1. Find the asymptote of this hyperbola.
Solution:
Using the one of the hyperbola formulas (for finding asymptotes):
y = y_{0 }− (b / a)x + (b / a)x_{0} and y = y_{0 }− (b / a)x + (b / a)x_{0}y = 2  ( 6/4)x + (6/4)5 and y = 2 + ( 6/4)x  (6/4)5
Answer: Asymptotes are y = 2  ( 3/2)x + (3/2)5, and y = 2 + ( 3/2)x  (3/2)5.
FAQs on Hyperbola
What is Hyperbola in Conic Section?
A hyperbola is the locus of a point whose difference of the distances from two fixed points is a constant value. The two fixed points are called the foci of the hyperbola, and the equation of the hyperbola is \(\dfrac{x^2}{a^2}  \dfrac{y^2}{b^2} = 1\). Here a is called the semimajor axis and b is called the semiminor axis of the hyperbola.
What Is the Equation of Hyperbola?
The equation of the hyperbola is \(\dfrac{x^2}{a^2}  \dfrac{y^2}{b^2} = 1\). Here 'a' is the semmajor axis, and 'b' is the semiminor axis. There are two standard forms of equations of a hyperbola.
How to Find the Equation of a Hyperbola?
The equation of the hyperbola can be derived from the basic definition of a hyperbola: A hyperbola is the locus of a point whose difference of the distances from two fixed points is a constant value. Let the fixed point be P(x, y), the foci are F and F'. Then the condition is PF  PF' = 2a. The difference is taken from the farther focus, and then the nearer focus. This on further substitutions and simplification we have the equation of the hyperbola as \(\dfrac{x^2}{a^2}  \dfrac{y^2}{b^2} = 1\).
What Is a Rectangular Hyperbola?
The hyperbola having the major axis and the minor axis of equal length is called a rectangular hyperbola. Hence we have 2a = 2b, or a = b. The equation of the rectangular hyperbola is x^{2}  y^{2} = a^{2}.
What is the Eccentricity of Hyperbola?
The eccentricity of the hyperbola is greater than 1. (e > 1). The eccentricity is the ratio of the distance of the focus from the center of the ellipse, and the distance of the vertex from the center of the ellipse. The distance of the focus is 'c' units, and the distance of the vertex is 'a' units, and hence the eccentricity is e = c/a. Also here we have c^{2} = a^{2} + b^{2}.
What Is the Foci of a Hyperbola?
The hyperbola has two foci on either side of its center, and on its transverse axis. The hyperbola \(\dfrac{x^2}{a^2}  \dfrac{y^2}{b^2} = 1\) has two foci (c, 0), and (c, 0).
What Is the Conjugate Axis of a Hyperbola?
The axis line passing through the center of the hyperbola and perpendicular to its transverse axis is called the conjugate axis of the hyperbola. The conjugate axis of the hyperbola having the equation \(\dfrac{x^2}{a^2}  \dfrac{y^2}{b^2} = 1\) is the yaxis.
What are Asymptotes of Hyperbola?
The asymptotes are the lines that are parallel to the hyperbola and are assumed to meet the hyperbola at infinity. The equation of asymptotes of the hyperbola are y = bx/a, and y = bx/a. The equation of pair of asymptotes of the hyperbola is \(\dfrac{x^2}{a^2}  \dfrac{y^2}{b^2} = 0\).
What Are the Vertices of a Hyperbola?
The vertices of a hyperbola are the points where the hyperbola cuts its transverse axis. The hyperbola has only two vertices, and the vertices of the hyperbola \(\dfrac{x^2}{a^2}  \dfrac{y^2}{b^2} = 1\) is (a, 0), and (a, 0) respectively.
How to Find Transverse Axis of a Hyperbola?
The transverse axis of a hyperbola is a line passing through the center and the two foci of the hyperbola. A hyperbola with an equation \(\dfrac{x^2}{a^2}  \dfrac{y^2}{b^2} = 1\) had the xaxis as its transverse axis.