Question

An ellipse (discussed in detail in Section 10.4 ) is generated by the parametric equations \(x=a \cos t, y=b \sin t.\) If \(0 < a < b,\) then the long axis (or major axis) lies on the \(y\) -axis and the short axis (or minor axis) lies on the \(x\) -axis. If \(0 < b < a,\) the axes are reversed. The lengths of the axes in the \(x\) - and \(y\) -directions are \(2 a\) and \(2 b,\) respectively. Sketch the graph of the following ellipses. Specify an interval in t over which the entire curve is generated. $$x=4 \cos t, y=9 \sin t$$

Short answer

Answer: The values of a and b are 4 and 9, respectively. The length of the major axis is 18, and the length of the minor axis is 8.

Step-by-step solution

Determine the values of a and b

Comparing the given parametric equations to the general parametric equations of an ellipse, we can directly see that \(a = 4\) and \(b = 9\).

Identify the major and minor axes

Since \(a=4\) and \(b=9\), we have \(0<a<b\). This means that the major axis lies on the y-axis and the minor axis lies on the x-axis. The lengths of the axes in the x- and y-directions are \(2a = 8\) and \(2b = 18\), respectively.

Sketch the graph of the ellipse

To sketch the graph, we can plot some points for different values of \(t\) and then join them in a smooth curve. For example, for \(t=0\), we get \(x=4\cos(0)=4\) and \(y=9\sin(0)=0\). Similarly, for \(t=\frac{\pi}{2}\), we get \(x=4\cos\left(\frac{\pi}{2}\right)=0\) and \(y=9\sin\left(\frac{\pi}{2}\right)=9\). Following this process, we can plot a few points and join them to get the ellipse. The resulting graph will show an ellipse with major axis of length 18 along the y-axis and a minor axis of length 8 along the x-axis. The shape of the ellipse is elongated vertically because the major axis is longer than the minor axis.

Specify the interval in t over which the entire curve is generated

Since the ellipse is generated by the trigonometric functions sine and cosine, we can choose an interval \(t \in [0, 2\pi]\). In this interval, sine and cosine functions both complete a full period, and the ellipse will be traced out completely once.

Key concepts

Major and Minor Axes of an Ellipse

The major and minor axes are the longest and shortest diameters of an ellipse, respectively. They intersect at the center of the ellipse and are perpendicular to each other. The distance between the center and a vertex on a major or minor axis is known respectively as the semi-major axis, denoted by 'a', and the semi-minor axis, denoted by 'b'.

When the ellipse is oriented such that its major axis is vertical, the parametric equations will have the form

\begin{large}\(x = a \cos t, y = b \sin t\)\end{large}

where 'a' is less than 'b'. This means the ellipse is taller than it is wide. Conversely, if 'a' is greater than 'b', the major axis would be horizontal, indicating the ellipse is wider than it is tall. For the ellipse given by the parametric equations \(x=4 \cos t, y=9 \sin t\), the semi-major axis 'b' is 9 and the semi-minor axis 'a' is 4.

Graphing Ellipses

To graph an ellipse using its parametric equations, we plot points by choosing values for 't' and computing the corresponding 'x' and 'y' coordinates. For instance, by plugging in certain key angles for 't' such as 0, \(\frac{\pi}{2}\), \(\pi\), and \(\frac{3\pi}{2}\), we obtain points that lie on the axes, marking the extent of the ellipse in those directions.

In the graphing process, we start by plotting the vertices of the ellipse, located at distances 'a' and 'b' from the origin along the major and minor axes. As we fold in more values of 't', we get more points that can be smoothly connected to show the elliptical shape. It's crucial to ensure the plot captures the correct proportions of the major and minor axes to accurately represent the ellipse's shape.

Trigonometric Parametric Equations

Parametric equations allow us to express the coordinates of points on curves as functions of a parameter. For an ellipse, trigonometric parametric equations use the sine and cosine functions to define the locus of points that form the shape. This pair of equations, \(x=a \cos t\) and \(y=b \sin t\), captures the essence of the ellipse by utilizing the cyclic properties of sine and cosine.

The use of trigonometric functions in parametric form is particularly effective because these functions inherently possess the periodicity required to loop and trace the ellipse from its starting point back to the origin. As 't' varies from 0 to \(2\pi\), the point (x, y) travels once around the ellipse, ensuring a continuous and closed curve is formed.

Sketching Curves in Calculus

In calculus, sketching curves involves understanding the behavior of functions and their graphical representations. For ellipses, we focus on the path traced out as the parameter 't' changes. It requires consideration of key points, symmetry, and the directional movement (counter-clockwise or clockwise) of the curve as 't' increases.

When sketching parametric curves such as ellipses, we trace the position of a moving point whose coordinates are determined by the parameter 't'. This sketching process not only aids in visualizing the shape of the curve but also reinforces the dynamic relationship between the algebraic parametric equations and their geometric interpretations on the graph. Each complete cycle of 't' from 0 to \(2\pi\) generates the entire ellipse, after which the cycle repeats.