Eccentricity

The eccentricity of any curved shape characterizes its shape, regardless of its size. The four curves that get formed when a plane intersects with the double-napped cone are circle, ellipse, parabola, and hyperbola. Their features are categorized based on their shapes that are determined by an interesting factor called eccentricity. The circles have zero eccentricity and the parabolas have unit eccentricity. The ellipses and hyperbolas have varying eccentricities. Let us learn more in detail about calculating the eccentricities of the conic sections.

The eccentricity of conic sections is defined as the ratio of the distance from any point on the conic section to the focus to the perpendicular distance from that point to the nearest directrix. For any conic section, the eccentricity of a conic section is the distance of any point on the curve to its focus ÷ the distance of the same point to its directrix = a constant. This constant value is known as eccentricity, which is denoted by e. The eccentricity of a curved shape determines how round the shape is. The curvatures decrease as the eccentricity increases.

If the eccentricities are big, the curves are less. Thus we conclude that the curvatures of these conic sections decrease as their eccentricities increase.

• The eccentricity of a circle = 0
• The eccentricity of an ellipse = between 0 and 1
• The eccentricity of a parabola = 1
• The eccentricity of a hyperbola > 1
• The eccentricity of a line = infinity

The planets revolve around the earth in an elliptical orbit. The eccentricity of earth's orbit(e = 0.0167) is less compared to that of Mars(e=0.0935). The more the value of eccentricity moves away from zero, the shape looks less like a circle. While an ellipse and a hyperbola have two foci and two directrixes, a parabola has one focus and one directrix. Their eccentricity formulas are given in terms of their semimajor axis(a) and semi-minor axis(b), in the case of an ellipse and a = semi-transverse axis and b = semi-conjugate axis in the case of a hyperbola. The formula of eccentricity is given by

Eccentricity = Distance to the focus/ Distance to the directrix.

e = c/a

where,

• e = eccentricity
• c = distance from any point on the conic section to its focus
• a= distance from any point on the conic section to its directrix

An ellipse is the set of all points in a plane, where the sum of distances from two fixed points(foci) in the plane is constant. The eccentricity of an ellipse is the ratio of the distance from its center to either of its foci and to one of its vertices. e = c/a. We know that c = \(\sqrt{a^2-b^2}\)

If a > b, e = \(\dfrac{\sqrt{a^2-b^2}}{a}\)

If a < b, e = \(\dfrac{\sqrt{b^2-a^2}}{b}\)

Where a = semi-major axis

• b = semi-minor axis and
• c = distance from the center of the ellipse to either focus. The eccentricity of an ellipse is 0 ≤ e< 1.

The set of all the points in a plane that are equidistant from a fixed point (center) in the plane is called the circle. A circle is an ellipse in which both the foci coincide with its center. As the foci are at the same point, for a circle, the distance from the center to a focus is zero. This eccentricity gives the circle its round shape. Thus the eccentricity of any circle is 0.

A parabola is the set of all the points in a plane that are equidistant from a fixed line called the directrix and a fixed point called the focus. The locus of the moving point P forms the parabola, which occurs when the eccentricity e = 1. This gives the U shape to the parabola curve. Thus the eccentricity of a parabola is always 1.

Hyperbola is the set of all the points, the difference of whose distances from the two fixed points in the plane (foci) is a constant. In a hyperbola, 2a is the length of the transverse axis and 2b is the length of the conjugate axis. The distance between the two foci is 2c. Similar to the ellipse, the hyperbola has an eccentricity which is the ratio of the c to a. Since c ≥ a, the eccentricity is never less than 1. The eccentricity of the hyperbola is given by e = \(\dfrac{\sqrt{a^2+b^2}}{a}\). The distance between the two foci = 2ae.

Tips and Tricks on Eccentricity:

• The eccentricity of the conic sections determines their curvatures.
• The eccentricity of a circle is 0 and that of a parabola is 1.
• The varying eccentricities of ellipses and parabola are calculated using the formula e = c/a, where c = \(\sqrt{a^2+b^2}\), where a and b are the semi-axes for a hyperbola and c= \(\sqrt{a^2-b^2}\) in the case of ellipse.

☛ Also Check:

• Locus
• Equation of a circle

What is Eccentricity?

Eccentricity is the mathematical constant that is given for a conic section. It is the ratio of the distances from any point of the conic section to its focus to the same point to its corresponding directrix. The eccentricity of a conic section tells the measure of how much the curve deviates from being circular.

What is The Formula of Eccentricity?

The eccentricity of a conic section is the distance of any to its focus/ the distance of the same point to its directrix. The formula of eccentricity is e = c/a, where c = √(a²+b²) and

c = distance from any point on the conic section to its focus

a= distance from any point on the conic section to its directrix

What is the Eccentricity of an Ellipse?

The eccentricity of an ellipse ranges between 0 and 1. The eccentricity of an ellipse is the ratio between the distances from the center of the ellipse to one of the foci and to one of the vertices of the ellipse. If the eccentricity reaches 0, it becomes a circle and if it reaches 1, it becomes a parabola.

What is the Eccentricity of a Circle?

The eccentricity of a circle is always one.

What is the Eccentricity of a Parabola?

The eccentricity of a parabola is always one. The distance between any point and its focus and the perpendicular distance between the same point and the directrix is equal. Thus c = a. Hence eccentricity e = c/a results in one.

Why is Eccentricity of a Circle Zero?

The eccentricity of a circle is always zero because the foci of the circle coincide at the center.

What Happens When Eccentricity is 1?

When the curve of an eccentricity is 1, then it means the curve is a parabola.

How do You Find the Eccentricity of a Conic?

The eccentricity is found by finding the ratio of the distance between any point on the conic section to its focus to the perpendicular distance from the point to its directrix.

What is the Maximum Value that an Eccentricity Can Be?

The eccentricity of a hyperbola is always greater than 1. When the eccentricity reaches infinity, it is no longer a curve and it is a straight line.

What is the Minimum Value that an Eccentricity Can Be?

The minimum value of eccentricity is 0, like that of a circle.