Problem 37 - A Graphical Approach to College Algebra

Question

Identify the type of conic section consisting of the set of all points in the plane for which the sum of the distances from the points \((5,0)\) and \((-5,0)\) is 14

Step-by-Step Solution

Answer

The conic section is an ellipse.

Step 1: Understand the Problem

We have a set of points where the sum of the distances from two fixed points, called foci, is constant. The given foci are \((5, 0)\) and \((-5, 0)\). The sum of distances is 14.

Step 2: Identify Characteristics of a Conic

For a conic section where the sum of distances from two fixed points (foci) is constant, the shape is an ellipse. According to the definition of an ellipse, the sum of distances from any point on the ellipse to the two foci is constant.

Step 3: Recognize the Conic Section

Based on the definition identified in the previous step, if the sum of the distances from any point on the conic to two fixed points is constant, this conic is an ellipse.

Key Concepts

Ellipse

An ellipse is a fascinating shape that you often find in both mathematics and nature. It is the set of all points in a plane such that the sum of their distances to two fixed points, called the "foci," is constant. This special property makes an ellipse unique among other conic sections, such as parabola and hyperbola. Here are a few key characteristics of ellipses:

• Ellipses are usually oval-shaped.
• The longer axis is called the "major axis," and the shorter one is the "minor axis."
• The midpoint of the major axis is the "center" of the ellipse.

Understanding the concept of an ellipse is essential when dealing with problems related to orbit shapes, optics, and many real-world applications.
In our exercise, the definition of an ellipse perfectly describes the set of points where the sum of distances from two fixed points, \(5, 0\) and \(-5, 0\), is a constant value of 14.

Foci

The foci of an ellipse are two special points that help define its shape and properties. Unlike a circle, which has a single center point, an ellipse has these two distinct foci. Here’s what you need to know about the foci:

• The foci always lie along the major axis of the ellipse.
• The distance between the foci and any point on the ellipse is constant.
• If you can move the foci but keep their distance sum constant, the ellipse will change size, but still, remain an ellipse.

In the given exercise, the foci are at \(5, 0\) and \(-5, 0\). The sum of the distances from any point on the ellipse to these foci remains 14, maintaining the signature property of the ellipse.

Distance Formula

The distance formula is a fundamental tool in geometry used to find the distance between two points in a plane. It is given by the formula: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]Here’s why it is important in the context of conic sections and ellipses:

• It helps calculate the distances from any point on the ellipse to the foci.
• By ensuring the sum of distances remains constant, you can verify whether a given set of points form an ellipse.
• It’s a foundational concept necessary for understanding the geometric properties of any shape.

In our exercise, the distance formula can be used to ensure that for any point on the ellipse, the sum of its distances to the foci \(5, 0\) and \(-5, 0\) will always equal 14, corroborating the properties of an ellipse. This understanding highlights why the distance formula is indispensable when working with ellipses and other geometric constructs.