Understanding the Eccentricity of an Ellipse with Specific Foci and Minor Axis Distance

An ellipse is a geometric shape defined by its properties such as the distance between its foci, the lengths of its axes, and its eccentricity. This article explores the relationship between the distance between the foci and the minor axis of an ellipse and how to determine the eccentricity under these specific conditions.

Properties of an Ellipse

Before delving into the problem, it is important to understand the key properties of an ellipse:

Semi-major axis (a): The semi-major axis is the longest radius of the ellipse. Semi-minor axis (b): The semi-minor axis is the shortest radius of the ellipse. Foci: The foci are points inside the ellipse, each of which is equidistant from the ellipse's surface. Eccentricity (e): The eccentricity of the ellipse, which is a measure of how much the ellipse deviates from a circle (0 for a perfect circle).

Given Conditions and Setup

The problem states that the distance between the foci of the ellipse is equal to the length of its minor axis. In mathematical terms:

Distance between foci: 2c 2b Length of the minor axis: 2b

From this, we can simplify to:

2c 2b

c b

Determining the Relationship

The relationship between the foci (c), the semi-major axis (a), and the eccentricity (e) of an ellipse is given by:

c ae

Additionally, the length of the minor axis (2b) relates to the semi-major and semi-minor axes as follows:

2b 2a^2 - 2b^2

Simplifying this, we get:

c^2 a^2 - b^2

Given c b, we substitute into the equation:

b^2 a^2 - b^2

This can be rearranged to:

2b^2 a^2

From this, we can express:

a b√2

Calculating the Eccentricity

The eccentricity (e) is defined as:

e c / a

Substituting c b and a b√2, we get:

e b / (b√2)

Simplifying:

e 1 / √2

Thus, the eccentricity of the ellipse is:

boxed{1 / √2}

Conclusion

In conclusion, we have demonstrated that if the distance between the foci of an ellipse is equal to the length of its minor axis, the eccentricity of the ellipse is 1/√2. This result is derived by leveraging the fundamental properties and relationships of an ellipse, specifically the semi-major axis, semi-minor axis, and the relationship between these elements and the eccentricity.

Related Topics

For further reading and exploration, consider delving into the following topics:

Ellipse Definition and Properties: A comprehensive understanding of the basic properties of ellipses, including the definitions of major and minor axes, foci, and eccentricity. Geometric Shapes and Calculations: Techniques for solving problems involving the geometry of ellipses and other conic sections. Eccentricity in Elliptical Motion: The role of eccentricity in the physics of elliptical orbits, often studied in celestial mechanics.