Problem 35 - Precalculus: Mathematics for Calculus

Question

Graphing Ellipses Use a graphing device to graph the ellipse. $$\frac{x^{2}}{25}+\frac{y^{2}}{20}=1$$

Step-by-Step Solution

Answer

The ellipse has a horizontal orientation with a center at the origin, a major axis length of 10, and a minor axis length of \(4\sqrt{5}\).

Step 1: Identify the Standard Form of the Ellipse Equation

The given equation \(\frac{x^{2}}{25} + \frac{y^{2}}{20} = 1\) is already in the standard form of an ellipse equation \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\). Here, \(a^2 = 25\) and \(b^2 = 20\).

Step 2: Determine the Values of a and b

Given \(a^2 = 25\), we find \(a = \sqrt{25} = 5\). Given \(b^2 = 20\), we find \(b = \sqrt{20} = \sqrt{4 \cdot 5} = 2\sqrt{5}\).

Step 3: Determine Orientation of the Ellipse

Since \(a^2 > b^2\), the ellipse is horizontal. This means the major axis is along the x-axis.

Step 4: Identify the Center and Axes Lengths

The center of the ellipse is at the origin \((0,0)\). The length of the major axis is \(2a = 10\) and the length of the minor axis is \(2b = 4\sqrt{5}\).

Step 5: Plot the Ellipse Using the Center and Axes

From the center \((0,0)\), move \(5\) units left and right along the x-axis and \(2\sqrt{5}\) units up and down along the y-axis to mark the vertices. Sketch the ellipse through these points.

Step 6: Use Graphing Device to Verify

Input the equation into a graphing calculator or software to verify the plotted points and check the shape of the ellipse.

Key Concepts

Standard Form of Ellipse

Ellipses have a specific way to represent their equations known as the standard form. It helps swiftly interpret the geometry of the ellipse. The standard form of an ellipse's equation with a horizontal major axis is given by

• \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\)

Where \(a\) and \(b\) are the semi-major and semi-minor axes, respectively. If the major axis is vertical, the equation is flipped:

• \(\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1\)

The standard form makes it easier to identify and graph ellipses, and to determine their properties such as the center, axes lengths, and orientation just from looking at the equation.
The key is ensuring that the equation equals 1, helping you see the values of \(a\) and \(b\) directly.

Ellipse Equation

At the heart of graphing ellipses lies their equation. For the given ellipse,

• \(\frac{x^{2}}{25} + \frac{y^{2}}{20} = 1\)

This indicates an ellipse centered at the origin \((0, 0)\). Each term of the equation brings specific insights:

• The term \(\frac{x^2}{a^2}\) gives information about the width of the ellipse.
• The term \(\frac{y^2}{b^2}\) provides the height details.

The division by \(a^2\) and \(b^2\) normalizes the ellipse to fit perfectly within an aligned rectangular frame spanned by the major and minor axes.
Beginning with this form, you can calculate axes lengths, positions of the vertices, and importantly graph this mathematical gem.

Ellipse Orientation

The orientation of an ellipse tells us which way the longest diameter of the ellipse points.
For the ellipse equation given, \(a^2 = 25\) and \(b^2 = 20\). Because \(a^2 > b^2\), it means the major axis is along the x-axis, indicating a horizontal orientation.

• Horizontal ellipses appear wider than they are tall.
• Vertical ellipses are taller than they are wide; for these, \(b^2\) would be greater than \(a^2\).

Understanding the orientation helps visualize the shape and ensure correct sketching of the ellipse. It primarily hinges on comparing \(a^2\) with \(b^2\).

Ellipse Axes Lengths

Axes lengths describe how far an ellipse stretches in both dimensions, informing about its shape.
The axes lengths are essential properties of an ellipse. They simplify understanding of the ellipse's size. Fundamentally, the lengths come from:

• The major axis, oriented horizontally; its length is \(2a\).
• The minor axis, perpendicular to the major; its length is \(2b\).

For our example, with \(a = 5\) and \(b = \sqrt{20} = 2\sqrt{5}\), the major axis length is:

• \(2a = 10\)

and the minor axis length is

• \(2b = 4\sqrt{5}\)

These measurements allow us to correctly draft the ellipse on a graph, showing exactly how it will spread around its center. Pacing these lengths on graph paper provides precise plotting points, creating an accurate picture of the ellipse's layout.