Problem 14 - Precalculus with Limits

Question

In Exercises 11-18, find the standard form of the equation of the ellipse with the given characteristics and center at the origin. Vertices: \((0, \pm8); \quad\) foci: \((0, \pm4)\)

Step-by-Step Solution

Answer

The standard form of the equation of the ellipse is \(\frac{y^2}{64} + \frac{x^2}{48} = 1\).

Step 1: Identify Ellipse Orientation and Major Axis Length

Given the vertices \((0,\pm8)\), the major axis length is \(2a = 16\). This gives us \(a = 8\). Also, because the vertices are on the y-axis, we know the ellipse is vertical.

Step 2: Identify the Distance from Center to Foci

The given foci are at \((0,\pm4)\) indicating that the distance from the center to either focus is \(c = 4\). This is half the distance between the two foci.

Step 3: Calculate Minor Axis Length

Using the formula \(a^2 = b^2 + c^2\), we can find the length of the minor axis. Rearranging for \(b^2\) yields \(b^2 = a^2 - c^2 = 8^2 - 4^2 = 64 - 16 = 48\). Thus, the minor axis length is \(2b = 2\sqrt{48}=4\sqrt{3}\).

Step 4: Write the Standard form of the Ellipse

Since this is a vertical ellipse, the standard equation should be written as \(\frac{y^2}{a^2} + \frac{x^2}{b^2} = 1\). Substituting the values of \(a\) and \(b\) that we found earlier, this becomes \(\frac{y^2}{64} + \frac{x^2}{48} = 1\).

Key Concepts

Standard Form

The standard form of an ellipse is a mathematical equation that represents the shape and size of an ellipse with respect to the coordinate plane. An ellipse centered at the origin \(0,0\) can be expressed in two possible formats, depending on its orientation.

• If the ellipse is oriented horizontally (major axis along the x-axis), the standard form is: \(rac{x^2}{a^2} + rac{y^2}{b^2} = 1\)
• For a vertical orientation (major axis along the y-axis), the form is: \(rac{y^2}{a^2} + rac{x^2}{b^2} = 1\)

Remember, in each case, \(a\) represents the semi-major axis, while \(b\) represents the semi-minor axis. These terms are derived from the full lengths of the axes which are \(2a\) and \(2b\), respectively. The larger value always determines the major axis.
Understanding the type of ellipse can help you choose the correct standard form equation.

Major Axis

The major axis of an ellipse is the longest diameter that spans from one vertex across the ellipse to the opposite vertex. It essentially represents the 'width' or 'height' of the ellipse, depending on its orientation. Here, the major axis is always aligned with direction defined by its largest value.

• For example, if the vertices are \( (0, \pm a)\), then the major axis is vertical.
• The length of the major axis is given by \(2a\), where \(a\) is the distance from the center to one vertex.

This axis is crucial because it helps define the true shape of the ellipse and correctly identify its orientation on the coordinate plane.
In our given problem, the vertices \( (0, \pm 8)\) suggest that the ellipse is vertical with \(a = 8\), resulting in a major axis length of \(2a = 16\).

Minor Axis

The minor axis of an ellipse is the shortest diameter, stretching from one side of the ellipse to the other through the center, perpendicular to the major axis. It represents the shorter side of the ellipse and is oriented across the narrowest part of the shape.

• For an ellipse with a vertical major axis, the minor axis will be horizontal.
• The length of the minor axis is denoted by \(2b\), where \(b\) is the distance from the center to the ellipse's edge along this axis.

In solving for the minor axis, we use the relationship \(a^2 = b^2 + c^2\), derived from the ellipse's geometric properties. In this exercise, substituting the known values results in \(b^2 = 48\), giving a minor axis length of \(2b = 4\sqrt{3}\). This calculation is essential for writing the ellipse's complete standard equation.

Foci

The foci (singular: focus) of an ellipse are two special points located along the major axis and are crucial in defining the ellipse's shape. The distance between the center and each focus is represented by \(c\), and they help determine the ellipse's eccentricity, a measure of its deviation from being circular.

• The distance from the center to the foci, \(c\), is calculated using the formula \(a^2 = b^2 + c^2\).
• In this exercise, the foci are given at \( (0, \pm 4)\), indicating \(c = 4\).

These focal points play a key role in various properties and may be used to derive certain characteristics of the ellipse, such as its geometric symmetry and the reflective properties applicable in optics. The knowledge of the ellipse's foci helps finalize understanding of its full shape.