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 and minor axes of lengths 12 and \(2,\) on the \(x\) - and \(y\) -axes, respectively, generated clockwise

Short answer

Question: Write down the parametric equations for an ellipse centered at the origin with major and minor axes of length 12 and 2, respectively, generated clockwise and find its description in terms of x and y. Solution: Based on our analysis, the parametric equations of the given ellipse are: $$x = 12\cos t$$ $$y = -2\sin t$$ and the description of the ellipse in terms of x and y is: $$\frac{x^2}{144} + \frac{y^2}{4} = 1$$

Step-by-step solution

Write down the standard form of parametric equations

For an ellipse centered at the origin with major and minor axes on the x- and y-axes, the standard form of parametric equations is: $$x = a\cos t$$ $$y = b\sin t$$ where a and b are lengths of the major and minor axes respectively, and t is the parameter.

Substitute the given values of a and b into the equations

We are given that the lengths of the major and minor axes are 12 and 2 respectively. Therefore, we substitute these values into the equations and use negative t values for clockwise generation: $$x = 12\cos (-t)$$ $$y = 2\sin (-t)$$

Simplify the parametric equations

Using the properties of trigonometric functions, we can simplify the parametric equations: $$x = 12\cos t$$ $$y = -2\sin t$$ These are the parametric equations of the given ellipse.

Find the description in terms of x and y

To find the description of the ellipse in terms of x and y, we can eliminate the parameter t from the parametric equations. We can do this by solving for \(\cos^2 t\) and \(\sin^2 t\) from the equations: $$\cos^2 t = \frac{x^2}{12^2}$$ $$\sin^2 t = \frac{(-y)^2}{2^2}$$ Using the trigonometric identity \(\cos^2 t + \sin^2 t = 1\), we substitute and get: $$\frac{x^2}{12^2} + \frac{y^2}{2^2} = 1$$ This is the description of the ellipse in terms of x and y: $$\frac{x^2}{144} + \frac{y^2}{4} = 1$$

Graph the ellipse

To graph the ellipse, plot the points that satisfy the equation \(\frac{x^2}{144} + \frac{y^2}{4} = 1\) on the x-y plane. The major and minor axes are aligned with the x- and y-axes, so the shape of the ellipse should be elongated along the x-axis and short along the y-axis. The ellipse should also be symmetric with respect to both x and y axes. When graphed, the ellipse should look like this: [Graph link: https://www.desmos.com/calculator/rospwhzb5k]

Key concepts

Ellipses

An ellipse is a fascinating geometric shape that looks like a stretched-out circle. It's defined by its major and minor axes, which are the longest and shortest diameters, respectively.

The general equation for an ellipse centered at the origin is:

• In parametric form: \( x = a\cos t \), \( y = b\sin t \), where \(a\) and \(b\) are lengths of the major and minor axes, respectively.
• In Cartesian form: \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \).

These forms help characterize the ellipse's shape and position on the coordinate plane.

Ellipses are widely used in physics, astronomy, and engineering, where they can represent orbits, lenses, and more. Understanding ellipses' equations allows you to model real-life phenomena and provides insights into their properties.

Graphing Ellipses

Graphing an ellipse involves understanding its major and minor axes, as well as its orientation. In our example, the ellipse is centered at the origin (0,0), with the major axis along the x-axis and the minor axis along the y-axis.

Here’s a simple way to graph an ellipse:

• Identify the length of the major (12) and minor (2) axes.
• Draw axes using these two figures: spread 12 from the center along the x-axis and 2 along the y-axis.
• Plot key points at the ends of these axes: (12,0), (-12,0), (0,2), and (0,-2).
• Smoothly curve lines to connect these points, making sure the curve is elongated along the x-axis.

Note that an ellipse's graph will be symmetric about the x- and y-axes, meaning you can fold it over either axis and it'll match up perfectly.

Tools like Desmos or graphing calculators make it easy to visualize these shapes accurately, ensuring a better understanding of their properties and appearance.

Trigonometric Identities

Trigonometric identities are mathematical equations that relate the angles and sides of triangles, specifically right triangles. These are crucial when working with parametric equations of ellipses.

One of the key identities used is the Pythagorean identity:
\(\cos^2 t + \sin^2 t = 1.\)
This identity is handy because it helps transform parametric equations into Cartesian form.

When we started with the parametric equations for the ellipse \( x = a\cos t \) and \( y = b\sin t \), we used this identity to eliminate the parameter \( t \). By substituting \( \cos^2t = \frac{x^2}{a^2} \) and \( \sin^2t = \frac{y^2}{b^2} \) into the identity, we derived the Cartesian form of the ellipse.

Understanding these identities helps not just in ellipse equations, but also in solving various problems in physics and engineering that involve periodic phenomena.