Problem 80 - 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 (0,-4) with major and minor axes of lengths 10 and \(3,\) parallel to the \(x\) - and \(y\) -axes, respectively, generated clockwise (Hint: Shift the parametric equations.)

Step-by-Step Solution

Answer

Answer: The parametric equations of the ellipse are: \(x(t) = 10\sin{t}\) \(y(t) = -4 - 3\cos{t}\)

Step 1: Find the standard form equation of the ellipse

To find the standard form equation of the ellipse, we will use the following formula: \((x - h)^2 / a^2 + (y - k)^2 / b^2 = 1\) Here, (h, k) is the center of the ellipse, and a and b are the lengths of the major and minor axes, respectively. Given the center of the ellipse is (0, -4), the major axis has a length of 10, and the minor axis has a length of 3. So the equation becomes: \((x - 0)^2 / 10^2 + (y + 4)^2 / 3^2 = 1\) Simplifying the equation, we get: \(x^2 / 100 + (y + 4)^2 / 9 = 1\)

Step 2: Obtain the parametric equations of the ellipse

Given that the ellipse is generated clockwise, we can use the following parametric equations: \(x(t) = h + a\sin{t}\) \(y(t) = k - b\cos{t}\) Here, (h, k) is the center of the ellipse, a and b are the lengths of the major and minor axes, respectively, and t is the parameter. Substituting the given values (h, k) = (0, -4), a = 10, and b = 3, we get: \(x(t) = 0 + 10\sin{t} = 10\sin{t}\) \(y(t) = -4 - 3\cos{t}\) So the parametric equations will be: \(x(t) = 10\sin{t}\) \(y(t) = -4 - 3\cos{t}\)

Step 3: Graph the ellipse

To graph the ellipse, plot the parametric equations \(x(t) = 10\sin{t}\) and \(y(t) = -4 - 3\cos{t}\) on a coordinate plane. Since an ellipse is closed, plot the points for values of \(t\) between \(0\) and \(2\pi\). The graph should look like an ellipse centered at (0, -4), with the major and minor axes parallel to the x- and y-axes.

Step 4: Describe the ellipse in terms of x and y

Now that we have the standard form equation, parametric equations, and graph of the ellipse, we can describe it in terms of x and y. The standard form equation is: \(x^2 / 100 + (y + 4)^2 / 9 = 1\) This equation represents an ellipse centered at (0, -4) with a major axis length of 10 along the x-axis and a minor axis length of 3 along the y-axis. The ellipse is generated clockwise, making its parametric equations: \(x(t) = 10\sin{t}\) \(y(t) = -4 - 3\cos{t}\)

Key Concepts

Standard Form Equation of an Ellipse

Understanding the standard form equation of an ellipse provides a solid foundation for analyzing its geometric properties. In essence, the standard form is a specific representation of the ellipse that allows it to be graphed and understood algebraically. The formula is given by \[\begin{equation} \frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1 \end{equation}\]where

• \((h, k)\) represents the coordinates of the center of the ellipse,
• \(a\) is the length of the semi-major axis,
• \(b\) is the length of the semi-minor axis.

In the example provided, the standard form equation is \[\begin{equation} \frac{x^2}{100} + \frac{(y + 4)^2}{9} = 1 \end{equation}\]indicating our ellipse is centered at the point (0, -4), with a major axis of length 10 (hence, the semi-major axis is 5) and a minor axis of length 6 (semi-minor axis being 3). Note that the denominators of the equation, 100 and 9, are the squares of the lengths of the semi-axes. It should also be observed that when the values of \(x\) or \(y\) are set equal to the center coordinates, we get either the major or minor axis, indicating the tight link between the algebraic and geometric components of the ellipse.
By analyzing the standard form equation, one can determine the shape, location, and orientation of the ellipse with respect to its axes.

Graphing Ellipses in Polar Coordinates

When it comes to graphing ellipses, polar coordinates offer an alternative method to express points on a plane using distance from a reference point and an angle from a reference direction. Unfortunately, an ellipse does not have a straightforward representation in polar coordinates like a circle does. However, if one needs to graph an ellipse in polar coordinates, it should be done by converting the parametric equations or the Cartesian coordinates into polar coordinates — a process which involves trigonometry and might not result in a standard simple polar equation. For our ellipse centered at (0, -4) with a major axis length of 10 and a minor axis length of 3, we'd need to use the relationship between Cartesian and polar coordinates, where \(x = r\text{{sin}}(\theta)\) and \(y = r\text{{cos}}(\theta)\), and apply the necessary shifts and transformations to accurately depict the ellipse in a polar plane.

Describing Ellipses with Parametric Equations

Ensuring a comprehensive understanding of ellipses can further be enhanced by describing them through parametric equations. Parametric equations give each coordinate a separate expression, related by an independent parameter, usually denoted as \(t\text{—time}\). These equations are particularly useful when the shape to be described exhibits symmetry, as is the case with ellipses. For the ellipse in the example, which is centered at (0, -4) and has axes lengths of 10 (major) and 3 (minor), the parametric equations in terms of a counter-clockwise rotation are given as: \[\begin{equation} x(t) = 10\text{{sin}}(t) \end{equation}\] and\[\begin{equation} y(t) = -4 - 3\text{{cos}}(t) \end{equation}\]The parameter \(t\) traces the ellipse as it varies from 0 to \(2\text{{pi rad}}\), completing a full rotation around the ellipse. This parametric description is invaluable when needing to calculate specific points along the ellipse at given times or when working with motion and trajectories in physics. The sine and cosine functions ensure the x and y values correspond to points on our ellipse as they inherently satisfy the standard form equation when \(t\) is varied.