Eccentricity of Ellipse Formula, Definition, Derivation & Examples

The eccentricity of an ellipse is a measure of how much it deviates from being a perfect circle, with a value always less than 1. This parameter gives insight into the shape of the ellipse, indicating how elongated or "oval" it is. An eccentricity close to 1 suggests a highly elongated shape, whereas values closer to 0 indicate a shape that is nearly circular.

Eccentricity is defined as the ratio of the distances from any point on the ellipse to the focus and to the directrix. It helps in quantifying the deviation of the ellipse from circularity.

Definition and Formula

It is crucial to understand two fundamental terms: the focus and the directrix. In the context of conic sections, an ellipse is defined as the locus of points where the ratio of the distance from a fixed point (the focus) to a fixed line (the directrix) remains constant. This constant ratio is what we call the eccentricity.

Mathematically, for an ellipse, the eccentricity e is given by the formula:

e=c/a

where c is the distance from the center of the ellipse to the focus, and a is the distance from the center to one of the vertices along the major axis. Since e < 1 for an ellipse, it ensures that c is always less than a, reinforcing the ellipse's shape as more circular than elongated.

Discover about the Chapter video:

Conic Sections Detailed Video Explanation:

---

Formula for the Eccentricity of an Ellipse

The eccentricity of an ellipse is a value that always remains less than 1 (i.e., e<1). It provides a measure of the ellipse's shape, describing how much it deviates from being a perfect circle. Specifically, the eccentricity can be understood as the ratio of the distance from any point on the ellipse to its focus and the distance from the same point to the directrix.

Eccentricity of Ellipse

Mathematical Expression

\(e = \frac{\text{Distance from Focus}}{\text{Distance from Directrix}}\)

This formula encapsulates how the ellipse's shape stretches along its major axis compared to its distance from the fixed line (directrix). The closer the eccentricity is to 0, the more circular the ellipse; the closer it is to 1, the more elongated or oval it becomes.

e = c/a

Substituting the value of c we have the following value of eccentricity.

\(e = \sqrt{1- {b^2\over a^2}}\)

Here a is the length of the semi-major axis and b is the length of the semi-minor axis.

---

Derivation of Eccentricity of an Ellipse

To derive the eccentricity of an ellipse, we first need to establish the relationship between its semi-major axis (a), semi-minor axis (b), and the distance from the center to the focus (c). This relationship is crucial in understanding the geometric properties of an ellipse.

Derivation of Eccentricity of Ellipse

1. Define the Semi-Major and Semi-Minor Axes:
   • The length of the major axis is 2a.
   • The length of the minor axis is 2b.
   • The distance between the two foci is 2c.
2. Relationship between a, b, and c: By definition, for an ellipse, the sum of the distances from any point on the ellipse to the two foci is constant and equals the length of the major axis. Consider a point P on the ellipse at one end of the major axis.
3. Distance Calculation: Let F and F' be the foci of the ellipse. We know that for any point P on the ellipse:
   PF+PF′=2a
4. Consider Point P at One End of the Major Axis: Without loss of generality, let P be at (a,0) on the ellipse. Thus, the distances from P to the foci F and F' are given by a−c and a+c, respectively.
5. Sum of Distances from P to the Foci:
   PF+PF′=(a−c)+(a+c)=2a
6. Establishing the Relationship: Using the distance formula and the definition of an ellipse, we can derive the relationship between a, b and c:
   From the Pythagorean theorem applied to the right triangle formed by the semi-major axis, semi-minor axis, and the distance to the focus, we get:
   \(c^2 = a^2 - b^2\)
7. Eccentricity Formula: The eccentricity e is defined as:
   \(e = \frac{c}{a}\)

Substituting c from the relationship \(c^2 = a^2 - b^2\):

\(e = ​\frac {\sqrt{a^2 - b^2}} {a} \)

Thus, the eccentricity of an ellipse is derived as:

\(e = ​\frac {\sqrt{a^2 - b^2}} {a} \)

This formula expresses the eccentricity in terms of the semi-major and semi-minor axes, indicating the ellipse's deviation from a perfect circle.

---

Solved Examples of Eccentricity of an Ellipse

Example 1: An ellipse with a semi-major axis a=10 and a semi-minor axis b=8. Find the eccentricity of the ellipse.

Solution: Calculate c (distance from the center to the focus):
\(c^2=a^2−b^2\)

\(c^2 =10^2−8^2\)

\(c^2 =100−64\)

\(c^2=36\)

\(c =6\)

Calculate the eccentricity e:

\(e = \frac{c}{a} \)

\(e = \frac{6}{10} \)

So, the eccentricity of the ellipse is 0.60.

Example 2: An ellipse with a semi-major axis a=7 and an eccentricity e=0.5. Find the length of the semi-minor axis b.

Solution: Calculate c using the given eccentricity:
\(e = \frac{c}{a}  ⟹  c=e⋅a\)

\(=0.5⋅7 = 3.5\)

Calculate b using the relationship \(c^2 = a^2 - b^2\):

\(\implies b^2 = a^2 - c^2 \)

\(\implies b^2 = 7^2 - 3.5^2 = 49 - 12.25 = 36.75\)

\(\implies b = \sqrt{36.75} \approx 6.06\)

So, the length of the semi-minor axis b is approximately 6.0.

Example 3: An ellipse with a semi-major axis a=12 and a focus at c=9. Find the eccentricity and the length of the semi-minor axis b.

Solution: Calculate the eccentricity e:
\(e = \frac{c}{a} = \frac{9}{12} = 0.75e=ac=129=0.75\)

Calculate b using the relationship \(c^2 = a^2 - b^2\):
\(\implies b^2 = a^2 - c^2 \)

\(\implies b^2 = 12^2 - 9^2 \)

\(\implies 144 - 81 = 63\)

\( \implies b = \sqrt{63} \approx 7.94\)

So, the eccentricity of the ellipse is 0.75 and the length of the semi-minor axis b is approximately 7.94.