Problem 1 - College Algebra with Corequisite Support

Question

Define an ellipse in terms of its foci.

Step-by-Step Solution

Answer

An ellipse is defined by the set of points where the sum of distances from two fixed points (foci) is constant.

Step 1: Understanding the definition of an ellipse

An ellipse is a set of points such that the sum of the distances from two fixed points, called foci, is constant.

Step 2: Locating the foci

Assume two fixed points, F1 and F2, on a plane, these are the foci of the ellipse.

Step 3: Describing the constant sum

For any point P on the ellipse, the sum of the distances to F1 and F2 is constant. Mathematically, if P is any point on the ellipse, then \( PF1 + PF2 = 2a \), where \( a \) is the semi-major axis of the ellipse.

Step 4: Visualizing the ellipse

By varying point P while maintaining the condition \( PF1 + PF2 = 2a \), the locus of P forms the ellipse. The longest diameter of the ellipse is the major axis, and its length is \( 2a \).

Key Concepts

Foci

In an ellipse, the two fixed points known as foci (singular: focus) play a crucial role in defining its shape. An ellipse can be thought of as a stretched circle, and it is defined by the property that the sum of the distances from any point on the ellipse to these two foci is always constant.
This concept is beautifully illustrated by taking a piece of string, pinning the string ends to two focus points, and tracing a path with a pencil while keeping the string taut.

• The foci are denoted as F1 and F2 in mathematical equations.
• For any point P on the ellipse, the combined distance from P to both foci is consistent, creating the essential property of ellipses, noted as: \( PF1 + PF2 = 2a \).

Understanding the role of foci helps in determining the balance and dimensions of the ellipse, making them central to the study of its properties.

Semi-major axis

In ellipse geometry, the semi-major axis is a fundamental component. It is the longest radius from the center of the ellipse to its outer edge and serves as a dominant factor in shaping the ellipse.
An ellipse is often described in terms of its semi-major axis, which we denote as \( a \). The total length of the longest line that runs through the center and touch both ends of the ellipse is twice this length, or \( 2a \).

• The length of the semi-major axis directly influences the overall size of the ellipse, much like the radius affects the size of a circle.
• It provides a reference measure for other characteristics of an ellipse, including the distance between the foci.

The semi-major axis is fundamental to the formula \( PF1 + PF2 = 2a \), linking it to critical properties and measurements of the ellipse.

Loci

The concept of loci, in terms of an ellipse, refers to the path traced by a moving point that maintains a specific condition. In this case, the geometric condition is that the sum of the distances from this moving point to each of the foci remains constant.
In simpler terms, the locus is the collection of all points P that satisfy the equation \( PF1 + PF2 = 2a \), where F1 and F2 are fixed.

• The term 'locus' essentially means location, making it a perfect term for this collection of points.
• Each point on this path is equidistantly balanced concerning the two foci at any given moment.

This characteristic makes the ellipse a unique shape where understanding loci helps clarify the constant distance relationship intrinsic to ellipses.