Question

Find parametric equations (not unique) of the following ellipses (see Exercises \(71-72\) ). Graph the ellipse and find a description in terms of \(x\) and \(y\) An ellipse centered at the origin with major axis of length 6 on the \(x\) -axis and minor axis of length 3 on the \(y\) -axis, generated counterclockwise

Short answer

Answer: The parametric equations of the ellipse are: \(x = 3 \cos{t}\) \(y = 1.5 \sin{t}\) The cartesian equation of the ellipse is: \(\frac{x^2}{9} + \frac{y^2}{2.25} = 1\) The key features of this ellipse are: - Center at the origin (0, 0) - Major axis of length 6 lying on the x-axis - Minor axis of length 3 lying on the y-axis - Vertices at (3, 0), (-3, 0), (0, 1.5), and (0, -1.5)

Step-by-step solution

Find the parametric equations of the ellipse

To determine the parametric equations of the ellipse, first find the lengths of the semi-major and semi-minor axes. In this case, the semi-major axis a is half the length of the major axis (6 units), so a = 3. Likewise, the semi-minor axis b is half the length of the minor axis (3 units), so b = 1.5. The parametric equations of an ellipse centered at the origin are given by: \(x = a \cos{t}\) \(y = b \sin{t}\) where t is the parameter, ranging from 0 to \(2\pi\). In our case: \(x = 3 \cos{t}\) \(y = 1.5 \sin{t}\)

Transform the parametric equations into cartesian equations

We can now express the ellipse in the cartesian coordinate system in terms of x and y. To do this, we can write sin(t) and cos(t) in terms of x and y using the parametric equations: \(\cos{t} = \frac{x}{3}\) \(\sin{t} = \frac{y}{1.5}\) Now, we use the trigonometric identity: \(\sin^2{t} + \cos^2{t} = 1\) Substituting sin(t) and cos(t) with their expressions in x and y, we get: \(\left(\frac{x}{3}\right)^2 + \left(\frac{y}{1.5}\right)^2 = 1\) or more simply, \(\frac{x^2}{9} + \frac{y^2}{2.25} = 1\)

Graph the ellipse

To graph the ellipse, consider the Cartesian (x, y) equation we found in step 2 as: \(\frac{x^2}{9} + \frac{y^2}{2.25} = 1\) It can be observed that the maximal value of x is \(\pm3\) (at t=0 or t=\(\pi\)) and the maximal value of y is \(\pm1.5\) (at t=\(\frac{\pi}{2}\) or t=\(\frac{3\pi}{2}\)). With the standard equation for the ellipse and these coordinates of the vertices, you can graph the ellipse on the Cartesian plane. To complete the graph, draw the vertices at (3, 0), (-3, 0), (0, 1.5), and (0, -1.5), then plot the ellipse using the given equation: \(\frac{x^2}{9} + \frac{y^2}{2.25} = 1\) The ellipse will be elongated horizontally, with a major axis of length 6 lying on the x-axis and a minor axis of length 3 lying on the y-axis.

Key concepts

Ellipse

An ellipse is an interesting shape in geometry that looks like a stretched or squeezed circle. You can think of it as an oval. Mathematically, an ellipse is defined as the set of all points for which the sum of the distances to two specific points, known as foci, is constant. This unique property distinguishes ellipses from other conic sections like parabolas and hyperbolas.

In the context of the problem here, the given ellipse is centered at the origin. This means that its center is located at the point (0, 0) in the coordinate plane. The axes of the ellipse determine its shape and size, being the longest diameter known as the major axis and the shortest diameter called the minor axis.

Ellipses are fundamental in mathematics, appearing in various scientific fields like astronomy, where they describe the orbits of planets. To fully understand an ellipse, we often use a combination of Cartesian coordinates and parametric equations.

Cartesian Coordinates

Cartesian coordinates are a way to determine the positions of points on a plane using two numbers: \(x\) and \(y\). Think of them as the same coordinate grid you use on a map. The first number, \(x\), tells you how far to travel horizontally from the center of the graph, known as the origin. The second number, \(y\), tells you how far to move vertically.

The given ellipse in the problem is described in Cartesian coordinates. This involves transforming its parametric equations into an equation relating \(x\) and \(y\). The transformation makes it easier to analyze and graph the ellipse since it provides a clear visual representation of its shape on the coordinate plane.

Using Cartesian coordinates also allows for simple calculations and transformations, making them an essential tool in both theoretical studies and practical applications.

Trigonometric Identity

Trigonometric identities are equations involving trigonometric functions that are true for any value of the variables used. In this exercise, we used the famous Pythagorean identity: \( \sin^2{t} + \cos^2{t} = 1 \). This identity is a fundamental concept in trigonometry, showing the inherent relationship between the sine and cosine functions.

To understand the ellipse better, we expressed \(\cos{t}\) and \(\sin{t}\) in terms of the \(x\) and \(y\) coordinates of the ellipse. By substituting these into the trigonometric identity, we derived the Cartesian equation of the ellipse:
\[ \left(\frac{x}{3}\right)^2 + \left(\frac{y}{1.5}\right)^2 = 1 \]
This equation reveals the ellipse's symmetry and confirms its geometric properties using trigonometry, highlighting the beauty and utility of trigonometric identities in mathematical analysis.

Semi-Major Axis

A semi-major axis is half of the major axis of an ellipse. It is the longest radius that extends from the center of the ellipse to its edge along the longest dimension. In this problem, the major axis of the ellipse is 6 units long, so the semi-major axis, \(a\), is \(3\) units.

The semi-major axis is a critical component in defining the ellipse's size and orientation. It helps in writing the parametric equations of the ellipse. For instance, the equation \(x = a \cos{t}\) uses the length of the semi-major axis to establish the horizontal distance from the center for any angle \(t\).

Knowing the semi-major axis also assists in graphing the ellipse. It tells you how far the ellipse stretches horizontally from the center, allowing you to plot it accurately on a coordinate plane.

Semi-Minor Axis

The semi-minor axis is half of the minor axis of an ellipse, representing the shortest radius from its center to its edge. In the example provided, the minor axis measures 3 units, making the semi-minor axis, \(b\), equal to \(1.5\) units.

The semi-minor axis determines how much the ellipse narrows as you move away from the center along the shorter dimension. It plays a crucial role in the parametric equations of the ellipse, specifically in the vertical distance equation \(y = b \sin{t}\). This equation uses the semi-minor axis's length to track the vertical extent of the ellipse as the angle \(t\) changes.

Graphically, the semi-minor axis helps visualize the ellipse's height. While the semi-major axis tells you about the ellipse's width, the semi-minor axis outlines its height, creating an accurate representation of the ellipse's overall shape.