Problem 30 - 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. Eccentricity \(\frac{4}{7}\) vertices \(V \pm 7,0)\)

Step-by-Step Solution

Answer

Equation: \( \frac{x^2}{49} + \frac{y^2}{33} = 1 \).

Step 1: Identify the Standard Form of the Ellipse Equation

For an ellipse centered at the origin with a horizontal major axis, the standard form of the ellipse equation is \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \). Here, \(a\) is the semi-major axis and \(b\) is the semi-minor axis.

Step 2: Determine the Value of \(a\)

The vertices are given as \( (\pm 7, 0) \). The distance from the center to a vertex on the major axis is \(a = 7\).

Step 3: Calculate the Eccentricity Formula

The eccentricity \(e\) for an ellipse is given by \( e = \frac{c}{a} \), where \(c\) is the distance from the center to a focus. Since \( e = \frac{4}{7} \), we have \( \frac{c}{7} = \frac{4}{7} \).

Step 4: Solve for \(c\)

Simplifying \( \frac{c}{7} = \frac{4}{7} \) gives \(c = 4\). This means each focus is 4 units away from the center along the x-axis.

Step 5: Use the Relationship \(c^2 = a^2 - b^2\) to Find \(b^2\)

Using the relationship for ellipses \( c^2 = a^2 - b^2 \), substitute the known values: \( 4^2 = 7^2 - b^2 \), which simplifies to \( 16 = 49 - b^2 \).

Step 6: Calculate \(b^2\)

Solving for \(b^2\), we have \( b^2 = 49 - 16 = 33 \).

Step 7: Write the Equation of the Ellipse

Substitute \(a^2 = 49\) and \(b^2 = 33\) into the standard form of the ellipse equation: \( \frac{x^2}{49} + \frac{y^2}{33} = 1 \).

Key Concepts

Eccentricity

Understanding the eccentricity of an ellipse is crucial to grasping its structure. Eccentricity, denoted by the letter \( e \), is a measure of how much an ellipse deviates from being a circle. It defines the shape of the ellipse:

• If \( e = 0 \), the ellipse is a circle.
• If \( 0 < e < 1 \), the ellipse ranges from a stretched circle to a very elongated shape.

In our exercise, the eccentricity is given as \( \frac{4}{7} \), indicating that this particular ellipse is moderately elongated. Remember, eccentricity is calculated as \( e = \frac{c}{a} \), where \( c \) is the distance from the center to a focus, and \( a \) is the length of the semi-major axis. For a detailed understanding, both \( c \) and \( a \) values need to be determined to explain the ellipse's characteristics.

Semi-major Axis

The semi-major axis is a fundamental component in defining an ellipse. It's the longest radius running from the center to the edge, aligning with the major axis of the ellipse. In the standard formula, this is signified by \( a \).
In the example problem, the vertices are \((\pm 7, 0)\), indicating the semi-major axis runs horizontally along the x-axis. Therefore, \( a = 7 \). This length helps define not only the size of the ellipse but also plays a role in the formula \( c^2 = a^2 - b^2 \) used to calculate the semi-minor axis.

Semi-minor Axis

In an ellipse, the semi-minor axis is the radius perpendicular to the semi-major axis that stretches from the center to the ellipse's edge at the shortest point. It's denoted as \( b \) in the ellipse equation.
Using the formula \( c^2 = a^2 - b^2 \), we can solve for \( b \) when the values of \( a \) and \( c \) are known. In the example provided, \( a = 7 \) and \( c = 4 \), allowing us to find that \( b^2 = 33 \). Therefore, \( b = \sqrt{33} \). Understanding the role of the semi-minor axis is key in visualizing the ellipse's squashed form.

Vertices

Vertices in an ellipse are the points on the curve that are the furthest away from the center and lie on the major axis. For a more mathematical view, they represent the endpoints of the semi-major axis. In our exercise, the vertices are given as \((\pm 7, 0)\), showing that they lie along the x-axis since the ellipse's major axis is horizontal.
This information is not only used to define \( a \) but also to set the boundaries of the equation's graph. The vertices are integral in sketching an ellipse since they highlight the maximum stretch along the major axis, giving the ellipse its characteristic elongated form.