Problem 42 - Precalculus: Functions and Graphs

Question

Exer. \(41-44:\) Find an equation for the set of points in an Xy-plane such that the sum of the distances from \(F\) and \(F\) is \(k\) $$F(12,0), \quad F(-12,0) ; \quad k=26$$

Step-by-Step Solution

Answer

The equation is \( \frac{x^2}{169} + \frac{y^2}{25} = 1 \).

Step 1: Understand the Problem

We need to find an equation for a set of points where the sum of distances from two fixed points (foci) is constant. This problem is asking us to find the equation of an ellipse.

Step 2: Define the Properties of the Ellipse

An ellipse can be defined as the set of all points where the sum of the distances from any point on the ellipse to two fixed points (foci) is constant. Here, the foci are given as \( F(12, 0) \) and \( F(-12, 0) \), and the constant \( k \) is given as 26.

Step 3: Use the Ellipse Standard Form Equation

The standard equation of an ellipse is \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \). We need to find \( a \) and \( b \) such that the sum of distances is 26 and the distance between the foci is 24 (which is \(2c\)).

Step 4: Calculate the Semi-Major Axis

For the given sum of distances \( k = 26 \), the length of the major axis \( 2a \) is equal to \( k \). Thus \( a = \frac{k}{2} = \frac{26}{2} = 13 \).

Step 5: Find the Value of c

The distance between the foci is \( 2c \), so \( c = \frac{12 - (-12)}{2} = 12 \).

Step 6: Relate c and a to Find b

In an ellipse, \( c^2 = a^2 - b^2 \). With \( c = 12 \) and \( a = 13 \), we have \( 12^2 = 13^2 - b^2 \). Solving for \( b \), we get \( b^2 = 13^2 - 12^2 = 169 - 144 = 25 \), thus \( b = 5 \).

Step 7: Write the Equation of the Ellipse

Using the values of \( a \) and \( b \), the equation of the ellipse is \( \frac{x^2}{13^2} + \frac{y^2}{5^2} = 1 \), or simplified, \( \frac{x^2}{169} + \frac{y^2}{25} = 1 \).

Key Concepts

Foci

In the context of an ellipse, the term 'foci' refers to two special fixed points that are used in the formal definition of the shape. These points are denoted typically as

• F1 and F2
• equidistant from the center of the ellipse along the major axis

To understand their importance, imagine any point on the boundary of an ellipse. The sum of the distances from this point to both foci remains constant, which is precisely what defines an elliptical shape.
In our exercise, the foci are located at

• F(12,0) and F(-12,0), positioned symmetrically on the x-axis

This symmetry simplifies calculations and plays a crucial role when deriving the equation of the ellipse.
Knowing the foci, it's easier to visualize the ellipse and comprehend how any point on it maintains the property of distance constancy from these fixed markers.

Ellipse Properties

An ellipse is a fascinating geometric shape characterized by a few key properties that make it distinct from other curves. Here are some important aspects to understand:

• Major and Minor Axes: The major axis is the longest span across the ellipse, and the minor axis is the shortest. The major axis always passes through both foci.
• The Center: Located at the midpoint of the segment connecting the foci. For our specific ellipse, this point is at the origin (0,0).
• Eccentricity: A measure of how "stretched" an ellipse is. It is defined as the ratio of the distance between foci (2c) to the length of the major axis (2a).

For an ellipse defined by its equation \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \), 'a' represents the semi-major axis and 'b' the semi-minor axis. Our example has its semi-major axis length at 13, and semi-minor axis at 5.
These properties help in visualizing the ellipse and predicting how it interacts with its plane, proving useful in everything from architectural designs to astronomical calculations.

Sum of Distances

The defining trait of an ellipse is the sum of distances from any point on it to its two foci.

• For an ellipse, this sum always stays constant

This distinct property is what distinguishes an ellipse from a circle, where any point on the circle has equal distance from a single center point.
In our problem, this sum of distances is defined to be 26. This means if you take any point on the ellipse and measure its distance to each focus, the total will always be 26.
This characteristic provides a method to derive the dimensions of an ellipse by setting distances equal to this constant value, forming the basis of the standard ellipse equation.