Problem 24 - Precalculus: Functions and Graphs

Question

Exer \(19-36:\) Find an equation for the ellipse that has its center at the origin and satisfies the given conditions. Foci \(F(0, \pm 4), \quad\) minor axis of length 4

Step-by-Step Solution

Answer

The ellipse is: \( \frac{x^2}{4} + \frac{y^2}{20} = 1 \).

Step 1: Understand the Definition of an Ellipse

An ellipse is defined as the set of points such that the sum of the distances from any point on the ellipse to two fixed points, called foci, is constant. The standard equation of an ellipse centered at the origin with a vertical major axis is \( \frac{x^2}{b^2} + \frac{y^2}{a^2} = 1 \), where \( a > b \).

Step 2: Identify the Given Elements

The foci of the ellipse are given as \( F(0, \pm 4) \). This means the distance \( c \) from the center to each focus is 4. Additionally, the minor axis, which is \( 2b \), is given as 4.

Step 3: Calculate Half the Minor Axis

Since the length of the minor axis is 4, we find \( b \) by dividing it by 2: \( b = \frac{4}{2} = 2 \).

Step 4: Use the Relationship Between \(a\), \(b\), and \(c\)

In an ellipse, \( c^2 = a^2 - b^2 \). We know \( c = 4 \) and \( b = 2 \), so we substitute these into the formula: \( 4^2 = a^2 - 2^2 \).

Step 5: Solve for \(a^2\)

Calculate \( a^2 \) by solving the equation: \( 16 = a^2 - 4 \). Therefore, \( a^2 = 20 \).

Step 6: Write the Equation of the Ellipse

Since \( a^2 = 20 \) and \( b^2 = 4 \), substitute these into the standard ellipse equation to get \( \frac{x^2}{4} + \frac{y^2}{20} = 1 \).

Key Concepts

Ellipse Definition

In simple terms, an ellipse is a squashed circle. It resembles an oval shape that can vary from being almost circular (when the two axes are close in length) to more elongated. The key geometric definition of an ellipse states that it comprises a set of points where the sum of the distances from two fixed points, known as the foci, is constant. If you imagine stretching a loop of string tied at both ends to the foci, the path traced by the pencil will outline an ellipse. This fundamental property differentiates ellipses from other geometrical shapes, such as circles, which have equidistant points from a single central point.

Ellipse Foci

The foci are two crucial points on an ellipse. In our task, the foci are located at the coordinates

• \( F(0, +4 ) \)
• \( F(0, -4) \)

This means each focus is 4 units away from the center of the ellipse, which is at the origin \((0,0)\). The distance from the center to any focus is denoted by \( c \). This distance is instrumental in determining the shape and equation of the ellipse. In typical scenarios, an ellipse's foci are aligned along the major axis, the longest diameter through the center. For a vertical ellipse, like the one in our problem, the foci lie along the y-axis.

Minor Axis

The minor axis of an ellipse is the shortest diameter and is perpendicular to the major axis. For our given ellipse, the minor axis length is 4 units. This information helps us find the value of \( b \), half the length of the minor axis. In mathematical terms:

• The total length of the minor axis = 4 units.
• Therefore, \( b = \frac{4}{2} = 2 \).

The minor axis plays a key role in forming the standard equation of the ellipse by being part of the equation denominator. It's crucial, along with the major axis, in defining how stretched the ellipse appears.

Standard Equation of an Ellipse

The standard equation of an ellipse with its center at the origin differs based on which axis is the major one. Since the major axis is vertical in our example, the equation takes the form: \[ \frac{x^2}{b^2} + \frac{y^2}{a^2} = 1 \] where:

• \( a \) is the semi-major axis length, or \( a^2 = 20 \) as calculated,
• \( b \) is the semi-minor axis length, or \( b^2 = 4 \),
• The formula \( c^2 = a^2 - b^2 \) is used to find the relationship between \( a \), \( b \), and the distance \( c \) to the foci.

Plugging our values, the equation for the ellipse becomes:\[ \frac{x^2}{4} + \frac{y^2}{20} = 1 \]This representation allows one to easily spot the axes lengths and positions of the foci, making it easier to analyze and graph the ellipse.