Problem 41 - 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 \(k\) $$F(3,0), \quad F(-3,0) ; \quad k=10$$

Step-by-Step Solution

Answer

The equation is \(\frac{x^2}{25} + \frac{y^2}{16} = 1\).

Step 1: Understanding the Problem

We need to find an equation for the set of points such that the sum of distances from any point on this set to two fixed points (foci) is constant. This is a standard problem involving ellipses.

Step 2: Identify Given Parameters

The foci are given as points \(F_1(3,0)\) and \(F_2(-3,0)\), and the constant sum of distances, \(k\), is 10.

Step 3: Relation to an Ellipse

For an ellipse, the sum of distances from any point on the ellipse to its two foci is equal to its major axis, which is given here as \(k=10\). This implies that the ellipse is centered at the origin since the foci are symmetrically located with respect to the y-axis.

Step 4: Calculate the Semi-Major Axis

The major axis length is \(2a=10\), thus \(a=5\). The semi-major axis, \(a\), is 5 along the x-axis between the center and the ends of the ellipse.

Step 5: Calculate the Semi-Minor Axis

The distance between the foci is \(2c = 6\), so \(c = 3\). Using the relationship \(c^2 = a^2 - b^2\), solve for \(b\). Substitute \(c = 3\) and \(a = 5\): \\(3^2 = 5^2 - b^2 \9 = 25 - b^2 \b^2 = 25 - 9 = 16 \b = 4\).

Step 6: Write the Ellipse Equation

The equation of an ellipse centered at the origin with semi-major axis \(a = 5\) and semi-minor axis \(b = 4\) is given by \\(\frac{x^2}{5^2} + \frac{y^2}{4^2} = 1 \\frac{x^2}{25} + \frac{y^2}{16} = 1.\)

Key Concepts

Foci

In an ellipse, the two fixed points known as the *foci* are crucial. These points determine the shape and equation of the ellipse. For our problem, the foci are given as \(F_1(3,0)\) and \(F_2(-3,0)\). Each point on the ellipse maintains a specific sum of distances from these foci, defining the ellipse's shape. These foci are always located on the ellipse's major axis, and their precise positioning determines the ellipse's eccentricity, or how stretched it appears.
A helpful way to visualize this is by imagining a piece of string attached to both foci. If you maintain the string taut with a pencil while moving it around, the path traced out is the ellipse. The total length of the string corresponds to the constant sum of distances from any point on the ellipse to the foci.

Semi-Major Axis

The semi-major axis is a vital component of an ellipse's structure. This is half of the longest diameter of the ellipse, stretching from the center to the furthest point on the edge.
Given the problem's setup, the semi-major axis is at the center of our ellipse at the origin. Its length is derived from the total distance formula, where it equals half the constant sum of the distances to the foci. For this problem, the sum of the distances is \(k = 10\), making the semi-major axis \(a = \frac{10}{2} = 5\).
The semi-major axis runs along the x-axis in our example, as the foci are positioned horizontally. It is vital in forming the equation of the ellipse, as it influences the elliptic curve's width.

Semi-Minor Axis

The semi-minor axis of an ellipse is perpendicular to the semi-major axis and can be thought of as half the shortest diameter of the ellipse.
To find its length, we use the relationship between the focal distance \(c\), the semi-major axis \(a\), and the semi-minor axis \(b\). This relationship is given by the equation \(c^2 = a^2 - b^2\). From our problem:

• \(a = 5\)
• \(c = 3\)

Substituting these values, we find \(b^2 = 25 - 9 = 16\), leading to \(b = 4\). Therefore, the semi-minor axis in this problem is 4 units long, which stretches along the y-axis.

Distance Formula

Understanding the distance formula is foundational when working with ellipses. This formula, generalized as \(d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}\), calculates the distance between two points in a coordinate plane.
In the context of ellipses, it helps determine the total distance from any point on the ellipse to each of the foci. In our problem,

• The sum of these distances remains constant, equaling \(k = 10\).

This consistent sum across the ellipse confirms the use of ellipse properties and ensures the accuracy of our calculated semi-major and semi-minor axes. Mastery of the distance formula is crucial, as it underlies these and many other geometric calculations.