Eccentricity of Hyperbola: Formula, Derivation and Solved Examples

The eccentricity is a characteristic that determines the geometry of any conic section. The ratio of the distances from the hyperbola's center to either of its vertices on each side of the focus is known as the eccentricity of the hyperbola. 

• Thus, the eccentricity of a hyperbola depends on the length of the conjugate axis and the transverse axis. 
• These lengths can be determined directly from the equation of hyperbola or by finding the coordinates of the foci and vertices. 
• The eccentricity of a hyperbola is always greater than unity i.e. e > 1. 
• This ratio of hyperbola helps us to examine how closely a hyperbola is in a circular shape, related to a circle. 
• It also determines the ovalness of the hyperbola.

Read More: Eccentricity of an Ellipse

Keyterms: Eccentricity, Hyperbola, conic section, geometry, conjugate axis, transverse axis, plane 

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Hyperbola

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The locus of all the points of a plane such that the difference between any two fixed points on the plane is constant is known as a hyperbola.

Hyperbola

• Here, the fixed points are called foci (represented by F₁ and F₂ in the figure) of the hyperbola. 
• F1 is the left focus with coordinate (-c, 0) and F₂ is the right focus with coordinate (c, 0). 
• The point P on the hyperbola has coordinates (x, y). The above figure represents the hyperbola if P₁F₂ - P₁F₁ = P₂F₂ - P₂F₁ = P₃F₁ - P₃F₂ = constant (k)
• The foci of the hyperbola are joined with the line segment then, the midpoint of this line segment is referred to as the center (O) of a hyperbola. 
• The line segment that passes through both the focus is the transverse axis.
• A conjugate axis of a hyperbola is the line axis that is parallel to the transverse axis.

The intersection point of the transverse axis with a hyperbola is represented by points A and B as shown above. From the above figure, 

Length of the Transverse axis = 2a

Length of the Conjugate axis = 2b

Length of two foci = 2c

The relationship between a,b, and c is given by;

b = √(c² – a²)

Discover about the Chapter video:

Conic Sections Detailed Video Explanation:

Read More: Tangent to a Circle

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Standard Equation of Hyperbola

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The equation of the hyperbola with a transverse axis along the x-axis and center at the origin is;

The equation of the hyperbola with a transverse axis along the y-axis and centered at the origin is; \(- \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\)

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Eccentricity of Hyperbola

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As discussed above, the eccentricity of a hyperbola is the ratio of the distance between the focus and center of the hyperbola to the distance of the either vertex from the center of the hyperbola.

Eccentricity of Hyperbola

From the figure, given the distance of focus from the center is ‘c’ and the distance of the vertex from the origin of a hyperbola is ‘a’. 

Then, the eccentricity (e) of the hyperbola is given by: c/a

The eccentricity of a hyperbola is always greater than 1, Thus, the condition e >1 always follows. 

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Eccentricity Formula of Hyperbola

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As discussed above, the formula for the eccentricity of a hyperbola in relation to the distance of focus and vertex from the center of the hyperbola. 

The eccentricity is given as;

e = c/a

Additionally, eccentricity is defined as the ratio of any point P on the hyperbola's distance from the focus to that point's distance from the directrix.

Directrix of Hyperbola: The directrix of Hyperbola is defined as a straight line used to generate a hyperbola. 

Then, according to the definition the eccentricity is given as;

Eccentricity (e) = Distance of Point P from Focus/ Distance of P from Directrix. 

e = c/a (where, c² = a² + b²)

Substituting the value of c (distance of focus from the center of the hyperbola), the formula of eccentricity will be;

\(e = \sqrt{1 – (b^2/a^2)}\)

where 'a' denotes the semi-major axis length and 'b' denotes the semi-minor axis length.

Read Also: Important Questions of Coordinate Geometry

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Derivation of Eccentricity of Hyperbola

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The eccentricity of a hyperbola can be derived from the equation of a hyperbola.

From the definition of hyperbola, it is the locus of points in the plane such that the difference in their distance from any two fixed points on the plane is constant. This difference is equal to the length of the transverse axis.

• Here two fixed points are the foci of the hyperbola and point P is the required point on the plane. 
• Then according to the definition a hyperbola with a transverse axis along the x-axis and the center at the origin. 
• Here the coordinate of the point P is , (x,y), and the coordinate of foci are F₁ (-c, 0) and F₂ (c, 0). 

PF₂ - PF₁ = 2a 

√[(x² + c²) + y²] - √[(x² - c²) + y²] = 2a (From Distance Formula)

√[(x² + c²) + y²] = 2a + √[(x² - c²) + y²]

Square the expression on both sides, the following expression;

(x² + c²) + y² = 4a² + (x² - c²) + y² + 4a√[(x² - c²) + y²]

Further simplifying the above equation, the equation of hyperbola

x²/a² - y²/(c² – a²) = 1

The above equation represents the hyperbola

Now, replace c² - a² = b² to get the simplified equation of hyperbola as

x²/a² - y²/(c² – a²) = 1

Further on simplifying,

c² = a² + b²

c² = √(a² + b²)

Replace this value of c in the expression of eccentricity, e = c/a to obtain the formula of the eccentricity of a hyperbola.

⇒ e = √(a² + b²)/ a

⇒ e = √(1 + b²/a²) given, a > b

For Conjugate Hyperbola

The hyperbola whose transverse axis is along the y-axis and conjugate axis along the x-axis. The equation of the conjugate hyperbola is given as, \(- \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\)

For the conjugate hyperbola b > a,

a² = b²(e² - 1)

Thus, the eccentricity formula is given as e = √(1 + a²/b²) given, b > a

Read Also: Distance Formula in Coordinate Geometry

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Things to Remember

• The eccentricity of a hyperbola is the ratio of the distance between the focus and center of the hyperbola to the distance of the either vertex from the center of the hyperbola.
• The eccentricity of a hyperbola is given as, e = c/a, where ‘c’ is the distance of focus and ‘a’ is the distance of the vertex from the center of the hyperbola. 
• The length ‘a’ is the preferred length of a semi-major axis and ‘b’ is the length of a semi-minor axis.
• For a hyperbola whose transverse axis is along the x-axis, a formula of eccentricity is given as e = √(1 + b²/a²) for a > b.
• For a hyperbola whose transverse axis is along the y-axis, a formula of eccentricity is given as e = √(1 + a²/b²) for b > a.
• The length of a semi-major axis is half the length of the Transverse axis (2a) and the Length of a semi-minor axis is half the length of a Conjugate axis (2b).
• The eccentricity of a rectangular hyperbola given by xy = c² is √2.
• The length of the latus rectum of the hyperbola is 2b²/a. Also, the Length of Latus Rectum (2a) = 2e x (distance of focus from directrix), where e is the eccentricity.

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