Problem 26 - Precalculus with Calculus Previews

Question

Graphically show the difference between the given curves. Assume that \(a>0\) and \(b>\) o, $$ \begin{array}{l} x=a \cos t, y=b \sin t, a>b, \pi \leq t \leq 2 \pi \\ x=a \sin t, y=b \cos t, a>b, \pi \leq t \leq 2 \pi \end{array} $$

Step-by-Step Solution

Answer

The first curve represents a horizontal ellipse top half, and the second represents a vertical ellipse top half, highlighting different orientations.

Step 1: Understand the Parametric Equations

The given parametric equations represent ellipses. For the first equation, \(x = a \cos t\) and \(y = b \sin t\) trace an ellipse where \(a\) is the semi-major axis and \(b\) is the semi-minor axis. Similarly, the second equation \(x = a \sin t\) and \(y = b \cos t\) also represent an ellipse but rotated by \(45^{\circ}\). We need to analyze the difference between these two representations from \(t = \pi\) to \(t = 2\pi\).

Step 2: Plot Each Parametric Curve

Plot the first curve for \(t\) in the range \(\pi \leq t \leq 2\pi\). The plotting should begin at \((x,y) = (a \cos \pi, 0) = (-a, 0)\) and end at \((a \cos 2\pi, 0) = (a, 0)\). For the second curve, begin plotting from \((x,y) = (a \sin \pi, b \cos \pi) = (0, -b)\) to \((a \sin 2\pi, b \cos 2\pi) = (0, b)\).

Step 3: Determine the Geometric Differences

Compare the geometric paths created by each set of parametric equations. The first curve describes the top half of an ellipse along the x-axis from \(-a\) to \(a\), while the second curve describes the top half of a vertically oriented ellipse along the y-axis from \(-b\) to \(b\). Because \(a > b\), the horizontal ellipse is wider, while the vertical ellipse has a narrower span.

Step 4: Contrast Between Horizontal and Vertical Orientations

Since \(a > b\), the first parametric set represents a horizontally stretched ellipse and the second represents a vertically stretched ellipse. From \(t = \pi\) to \(t = 2\pi\), these ellipses contrast their orientations along their respective major axes. The horizontal ellipse is centered along the x-axis, and the vertical ellipse is centered along the y-axis.

Key Concepts

Ellipses

An ellipse is a geometric shape that resembles a stretched or compressed circle. Unlike a circle, an ellipse has two axes: the major axis and the minor axis. These axes determine the extent of the stretch. In the case of parametric equations, an ellipse can be described by equations of the form \(x = a \cos t\) and \(y = b \sin t\), where \(a\) and \(b\) are constants. Here, \(a\) represents the semi-major axis, the longest radius, and \(b\) represents the semi-minor axis, the shortest radius. If \(a > b\), the ellipse is stretched more along the x-axis, making it wider. Conversely, if \(b > a\), it is stretched more along the y-axis, making it taller. Ellipses are fundamental in physics and engineering, often used to describe orbits and stress distributions. They reveal the relationship between different dimensions, allowing us to predict behavior and patterns in various systems.

Graphical Representation

Graphically representing an ellipse involves plotting its parametric equations over a range of values for \(t\). The value of \(t\) typically varies from \(0\) to \(2\pi\), completing a full rotation around the ellipse. However, how we interpret and draw the ellipse can be affected by the specified range. For example, in our exercise, the values of \(t\) range from \(\pi\) to \(2\pi\), dictating the representation of only the top half of the ellipse.
The first set of parametric equations given, \(x = a \cos t\) and \(y = b \sin t\), plots a horizontal ellipse from left to right, starting and ending on the x-axis. This suits a scenario where the semi-major axis \(a\) is longer than the semi-minor axis \(b\).

• The start point is \((-a, 0)\), equivalent to \(\cos \pi\).
• The end point is \((a, 0)\), equivalent to \(\cos 2\pi\).

Visualizing these points helps us understand the distribution and progression of the plotted points over the axis. Accurately plotting these ensures a correct depiction of the ellipse's shape and orientation.

Orientation

The term 'orientation' in the context of ellipses refers to how the ellipse is positioned relative to the coordinate axes. For instance, it can be either aligned with the x-axis to form a horizontal ellipse, or with the y-axis for a vertical ellipse. The distinction in orientation is essential to understanding the character of the geometric curves.
In our exercise, two different parametric equations produce ellipses with distinct orientations:

• A horizontally oriented ellipse is given by \(x = a \cos t\) and \(y = b \sin t\). Here, the ellipse extends wider along the x-axis.
• A vertically oriented ellipse forms with \(x = a \sin t\) and \(y = b \cos t\). In this case, the shape is taller and stretches along the y-axis.

Despite these differences, both sets of equations plot the top halves of potentially rotated versions of an ellipse, depending on their parameterization and range. Recognizing each ellipse's orientation informs our expectations of how different axes influence its spread.