Problem 78 - Calculus Early Transcendentals

Question

Find parametric equations (not unique) of the following ellipses (see Exercises \(75-76\) ). Graph the ellipse and find a description in terms of \(x\) and \(y.\) An ellipse centered at the origin with major and minor axes of lengths 12 and \(2,\) on the \(x\) - and \(y\) -axes, respectively, generated clockwise

Step-by-Step Solution

Answer

Question: Provide the parametric equations, graph, and description of an ellipse centered at the origin with a clockwise increasing angle, major axis length of 12, and minor axis length of 2. Answer: The parametric equations for the ellipse are \(x(t) = 12 \cos(-t)\) and \(y(t) = 2 \sin(-t)\). The ellipse is described by the standard equation \( \frac{x^2}{144} + \frac{y^2}{4} =1 \).

Step 1: Write the standard equation of the ellipse

To write the standard equation of the ellipse, first focus on the provided information: The ellipse is centered at the origin, with a major axis length of 12 and a minor axis length of 2. Therefore, we can write the ellipse equation as follows: \( \frac{x^2}{12^2} + \frac{y^2}{2^2} = 1 \)

Step 2: Find the parametric equations

Now that we have the standard equation, we can use it to find the parametric equations. We'll generate the parametric equations by substituting \(x\) and \(y\) with parametric expressions in terms of the angle parameter \(t\). In this case, the angle parameter \(t\) should have a clockwise increasing direction, so we substitute: \(x = 12 \cos(-t)\) and \(y = 2 \sin(-t)\) with the parametric equations becoming: \(x(t) = 12 \cos(-t)\) \(y(t) = 2 \sin(-t)\)

Step 3: Graph the ellipse and find a description in terms of x and y

To graph the ellipse, we can simply plot the parametric equations \(x(t) = 12 \cos(-t)\) and \(y(t) = 2 \sin(-t)\), yielding an ellipse centered at the origin with major and minor axes on the \(x\)- and \(y\)-axes, respectively, and axes lengths of 12 and 2. In terms of \(x\) and \(y\), the ellipse can be described using the standard equation derived in Step 1: \( \frac{x^2}{144} + \frac{y^2}{4} =1 \) Now that we have the parametric equations, graph, and description of the ellipse in terms of \(x\) and \(y\), we can conclude our solution.

Key Concepts

Ellipse

An ellipse is a distinctive geometrical shape resembling an elongated circle. It is defined as the set of all points where the sum of the distances from two fixed points, called foci, is a constant.
In an ellipse, the longest diameter is called the major axis, while the shortest is known as the minor axis. These axes intersect at the center of the ellipse.
An ellipse centered at the origin is often given special attention due to its simplicity and symmetry along the axes. When graphing such an ellipse, the foci are typically placed symmetrically relative to the center. This symmetry is useful in establishing a basic understanding of its properties, such as size and shape, which are described by the lengths of the major and minor axes.

Graphing

Graphing an ellipse involves mapping the set of all points that satisfy its equation onto a coordinate plane. This allows us to visualize the geometric shape and better understand its properties.
To graph an ellipse:

• Identify the center of the ellipse.
• Determine the lengths of the major and minor axes.
• Plot the endpoints of these axes from the center, both horizontally and vertically.
• Sketch the ellipse around these axes ensuring it’s symmetrical.

For example, an ellipse centered at the origin with a major axis of 12 and a minor axis of 2 can easily be sketched by plotting points (12, 0), (-12, 0), (0, 2), and (0, -2), then smoothly joining these to form the elliptical shape.

Standard Equation

The standard equation of an ellipse gives a clear mathematical description of its shape in terms of coordinates. An ellipse centered at the origin with axes aligned with the coordinate axes is expressed as: \[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\] where:

• \(a\) is the semi-major axis length.
• \(b\) is the semi-minor axis length.

This equation derives from the ellipse’s geometric properties and provides a straightforward way to analyze its features.
In the example provided, the ellipse has a major axis of length 12 and a minor axis of length 2, giving us the standard equation: \(\frac{x^2}{144} + \frac{y^2}{4} = 1\). This equation is essential in converting the ellipse from its algebraic description to a more visually and graphically interpretable form.

Parametric Representation

The parametric representation of an ellipse offers an alternative way to express its points using parameters. Unlike the standard form, which directly relates \(x\) and \(y\), the parametric form introduces a third variable, \(t\), typically representing an angle.
For an ellipse with semi-major and semi-minor axes \(a\) and \(b\), the parametric equations are:

• \(x(t) = a \cos(t)\)
• \(y(t) = b \sin(t)\)

These equations allow you to calculate the coordinates dynamically as \(t\) varies. When the ellipse is being generated clockwise, as stated in the exercise, the angle \(t\) is taken negatively, resulting in expressions like:\[x(t) = a \cos(-t)\] and \[y(t) = b \sin(-t)\].
This method of representation is quite practical for computer graphics and animations where control over direction and flow of points is required. It provides a versatile approach to describing the ellipse's geometry dynamically.