Problem 38 - Contemporary Precalculus

Question

Use the information given in Special Topics 10.3. A and summarized in the endpapers at the beginning of this book to find a parameterization of the conic section whose rectangular equation is given. Confirm your answer by graphing. $$\frac{(x-2)^{2}}{10}+\frac{(y+5)^{2}}{20}=1$$

Step-by-Step Solution

Answer

Question: Identify the type of conic section, find the lengths of the horizontal and vertical axes, and provide the parameterization of the given equation: \(\frac{(x-2)^{2}}{10}+\frac{(y+5)^{2}}{20}=1\) Answer: The conic section is an ellipse centered at (2, -5) with horizontal axis length \(\sqrt{10}\) and vertical axis length \(\sqrt{20}\). The parameterization of the ellipse is \((2 + \sqrt{10} \cos t, -5 + \sqrt{20} \sin t)\).

Step 1: Identify the type of conic section

To determine the type of conic section, we will look at the coefficients given in the equation and compare it to the standard equations for each type of conic. Analyzing the given equation: $$\frac{(x-2)^{2}}{10}+\frac{(y+5)^{2}}{20}=1$$ Since both squared terms are added and they have different coefficients, this represents an ellipse centered at (2, -5) with horizontal and vertical axes.

Step 2: Determine the horizontal and vertical axes lengths

To find the lengths of the horizontal and vertical axes, we will use the values of the coefficients in the denominator. $$a^2 = 10 \implies a = \sqrt{10}$$ $$b^2 = 20 \implies b = \sqrt{20}$$ The lengths of the horizontal and vertical axes are \(\sqrt{10}\) and \(\sqrt{20}\), respectively.

Step 3: Parameterize the ellipse

Now that we know the type of conic and the axes lengths, we can parameterize the conic section. For an ellipse with center (h, k) and axes lengths a and b, we can use the parametric equations: $$x = h + a \cos t$$ $$y = k + b \sin t$$ Plugging in the information from our problem, we get: $$x = 2 + \sqrt{10} \cos t$$ $$y = -5 + \sqrt{20} \sin t$$ So, the parameterization of the ellipse is \((2 + \sqrt{10} \cos t, -5 + \sqrt{20} \sin t)\).

Step 4: Graph the ellipse

To confirm the solution, we can graph the ellipse using the parameterized equations. As t varies from 0 to \(2\pi\), the ellipse will be traced out in a counterclockwise direction around the center (2, -5) with a horizontal axis of length \(\sqrt{10}\) and a vertical axis of length \(\sqrt{20}\). Plot the ellipse with the given information to confirm the solution.

Key Concepts

Ellipse

An ellipse is a type of conic section, easily identified by its oval shape. It is formed by slicing through a cone at an angle, which does not parallel the cone's base or extend through the base. An ellipse has two axes, a longer major axis and a shorter minor axis.
The characteristics of an ellipse can be identified from its equation:

• If you see two squared terms with positive coefficients and addition between them, it's likely an ellipse.
• The equation is typically in the form: \(\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1\), with \((h, k)\) representing the center.
• If \(a^2 > b^2\), the major axis is horizontal; if \(b^2 > a^2\), it's vertical.

In the given problem, \(\frac{(x-2)^2}{10} + \frac{(y+5)^2}{20} = 1\), you see that it's an ellipse with its center at \((2, -5)\). The different denominators (10 and 20) indicate a vertical major axis, as 20 is greater than 10.

Parametric Equations

Parametric equations present a way to express the coordinates of the points making up a geometric shape, like an ellipse, using a parameter, usually denoted by \(t\).
For an ellipse with a center \((h, k)\), and axis lengths \(a\) and \(b\), the standard parametric equations are:

• \(x = h + a \cos t\)
• \(y = k + b \sin t\)

This means we substitute our center point and axis lengths into these equations to describe the ellipse.
In the exercise a substitution provides the parametric form \(x = 2 + \sqrt{10} \cos t\) and \(y = -5 + \sqrt{20} \sin t\), showing the path traced by the ellipse as the parameter \(t\) changes from 0 to \(2\pi\). This transformation from the standard equation highlights the ellipse's directional flow, even creating boundaries for the ellipse in the x and y-coordinate space.

Graphing Conics

Graphing conics, like ellipses, can provide a visual confirmation of calculations. It is a vital tool in understanding their properties and structure.
The process generally involves:

• Identifying the type of conic section from the equation (e.g., ellipse, parabola).
• Finding the center, axes, and lengths from parameters \(a\) and \(b\).
• Using parametric equations to determine key points and the path of the conic.

For our given ellipse, \(\frac{(x-2)^2}{10} + \frac{(y+5)^2}{20} = 1\), the graph would center at \((2, -5)\) and extend \(\sqrt{10}\) units horizontally and \(\sqrt{20}\) units vertically. Graphing it by parameterizing helps visualize the trace of the ellipse as \(t\) varies, depicting it moving counterclockwise around the center.
The graph helps affirm not only the equation's parameters but also the geometric symmetry and proportionality, key concepts in the study of conic sections.