Exploring the Eccentricity of an Ellipse with Key Foci Distances

In the realm of geometric mathematics, the properties of ellipses, particularly their eccentricity, are of considerable interest. This article delves into the methodology of determining the eccentricity of an ellipse given specific foci and a known point, such as the origin. Through a detailed step-by-step approach, this guide will provide a clear understanding of the process and the underlying mathematical principles.

Introduction to Ellipses and Key Concepts

An ellipse is a conic section defined by the set of all points in a plane such that the sum of their distances from two fixed points (the foci) is constant. In an ellipse, the distance between the foci is crucial for understanding its shape and properties, including eccentricity. The eccentricity of an ellipse, denoted by e, is a measure of how elongated or flat the ellipse is, with a value between 0 and 1. When e approaches 0, the ellipse becomes more circular, and when e approaches 1, the ellipse becomes more elongated.

Determining Eccentricity: The Role of Foci and Origin

Suppose we are given that the foci of an ellipse are at points (512, 0) and (247, 0), and that the ellipse also passes through the origin (0, 0). The objective is to determine the eccentricity of this ellipse. This problem involves several key steps and mathematical relationships.

Step 1: Calculate the Distance Between Focii

The distance between the two foci (512, 0) and (247, 0) is given by the distance formula:

Distance |512 - 247| 265 units

Step 2: Relate Distance to Eccentricity

In an ellipse, the relationship between the distance between the foci (2c) and the semi-major axis (a) is given by the equation:

2c 265 → c 132.5

The eccentricity of the ellipse is defined as:

e c / a

Step 3: Find the Distance Between Origin and Second Focii

The origin point (0, 0) is known to lie on the ellipse. Using the ellipse equation and the fact that the ellipse passes through the origin, we can find the semi-major axis (a) and semi-minor axis (b). The general equation of a horizontal ellipse centered at the origin is:

(x^2 / a^2) (y^2 / b^2) 1

Since the ellipse passes through the origin (0, 0), substituting these coordinates into the ellipse equation gives us:

(0^2 / a^2) (0^2 / b^2) 1 → 0 0 1

This inherently means that the origin (0, 0) being on the ellipse does not provide direct information about a and b. However, we can use the known distance between the foci to establish the relationship between a and c and solve for e.

Step 4: Solve for Eccentricity

From the distance formula and the definition of eccentricity:

e c / a 132.5 / a

To find a, we can use the property that the sum of the distances from any point on the ellipse to the foci is constant and equals 2a. Since the ellipse passes through the origin, we know that the sum of the distances from the origin to both foci is 2a,

265 2a → a 132.5

Now we can calculate the eccentricity:

e 132.5 / 132.5 1

Therefore, the eccentricity of this ellipse is 1. However, this suggests a degenerate case where the ellipse would be a circle, which contradicts the given foci distances. Hence, we need to re-evaluate the steps and correct any assumptions.

Corrected Approach

Upon re-evaluating the problem, we realize that the correct approach is to use the relationship derived from the definition of an ellipse. Given that the origin (0, 0) is a point on the ellipse and the distance between the foci is 265 units, we can find the correct value of a by using the relationship:

(2c)^2 4a^2 - 4b^2

Simplifying, we get:

265^2 4a^2 - 4b^2

Since the ellipse passes through the origin, the sum of the distances from the origin to each focus is 2a, which is the major axis length:

265 2a → a 132.5

Using the relationship for eccentricity:

e c / a 132.5 / 132.5 1 - (b^2 / a^2)

Given that the ellipse passes through the origin, we can also use the distance formula:

c √(a^2 - b^2) √(132.5^2 - b^2) 132.5

Thus, the eccentricity of the ellipse is:

e 1 - (b^2 / 132.5^2)

This shows that the eccentricity e can be calculated based on the values of b and a.

Conclusion

Understanding the eccentricity of an ellipse through key foci distances and known points is crucial for many applications in mathematics and physics. By following a step-by-step approach and employing the correct mathematical relationships, we can accurately determine the eccentricity of an ellipse even with limited information. This method not only enhances our comprehension of geometric shapes but also provides a powerful tool for solving complex problems in various fields.

Additional Resources

To further explore the concept of eccentricity and ellipses, we recommend the following resources:

Wikipedia – Ellipse MathIsFun – Ellipse Properties Sequoia Capital – Ellipse Equations and Identities