Question

Parametric equations of ellipses Find parametric equations (not unique) of the following ellipses. Graph the ellipse and find a description in terms of \(x\) and \(y\). An ellipse centered at (-2,-3) with major and minor axes of lengths 30 and \(20,\) parallel to the \(x\) - and \(y\) -axes, respectively, generated counterclockwise (Hint: Shift the parametric equations.)

Short answer

Answer: The parametric equations for the given ellipse are x = -2 + 30cos(t) and y = -3 + 20sin(t), where 0 ≤ t ≤ 2π.

Step-by-step solution

Write the general equation of the ellipse

An ellipse with center \((h, k)\), major axis length \(2a\), minor axis length \(2b\) parallel to the x- and y-axes, respectively, has the following standard equation: \(\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1\)

Determine specific form of ellipse equation

In our case, the ellipse has its center at \((-2, -3)\), major axis length \(30\), and minor axis length \(20\). We can now substitute these values into the general equation: \(\frac{(x+2)^2}{30^2} + \frac{(y+3)^2}{20^2} = 1\)

Express ellipse equation in parametric form

In order to express the ellipse in parametric form, we will introduce a parameter \(t\): \(x = -2 + 30 \cos t\) \(y = -3 + 20 \sin t\) where \(0 \leq t \leq 2 \pi\).

Graph the ellipse and find a description in terms of x and y

To graph the ellipse and find a description in terms of \(x\) and \(y\), we can rewrite the parametric equations as follows: 1. Replace \(\cos t\) with \(\frac{x + 2}{30}\) and \(\sin t\) with \(\frac{y + 3}{20}\). 2. Square both equations: \((\frac{x + 2}{30})^2 = 1 - (\frac{y + 3}{20})^2\). 3. Solve for \(y\): \(y = -3 + 20 \sqrt{1 - (\frac{x + 2}{30})^2}\). The ellipse can now be described in terms of \(x\) and \(y\) as follows: \(y = -3 + 20 \sqrt{1 - (\frac{x + 2}{30})^2}\) Here is a graph of the ellipse with the given parametric equations: https://www.desmos.com/calculator/ogllmnikxm

Key concepts

Ellipse

An ellipse is a symmetric shape in coordinate geometry, appearing like an elongated circle. It is defined as the collection of points for which the sum of distances from two distinct points, called foci, is constant. Ellipses have two axes: the major axis, which is the longest diameter, and the minor axis, which is the shortest.

The standard equation of an ellipse centered at r>

• Center
• Axes aligned to the x- and y-axes, respectively

Given a center point

• (h, k), major axis
• a length of 2a, and minor axis
• a length of 2b

The equation is r>\[ \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \]. Thus, for our specific ellipse, it is r>\[ \frac{(x+2)^2}{30^2} + \frac{(y+3)^2}{20^2} = 1 \].

Understanding the basic form of ellipse equations is crucial for graphing and translating them into other forms.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is the study of geometry using a coordinate system. This allows for the expression of geometric figures, such as ellipses, circles, and lines, as algebraic equations. It merges algebra with geometry, revolutionizing how individual shapes are analyzed.

In coMMath language, ellipses can be expressed in terms of x and y coordinates.

Key concepts include:

• Using coordinates to represent points on a plane
• Identifying geometric shapes through algebraic equations
• Connecting geometry with algebraic approaches

In our ellipse example, the center is at (-2, -3), and the equations reflect translations and stretches along the coordinate axes.

Coordinate geometry helps visualize and calculate the properties of shapes like ellipses, making it an essential tool in mathematics.

Parametric Form

The parametric form of an equation uses one or more parameters to express the coordinates of points on a geometric object. In the case of an ellipse, parametric equations facilitate a different way to describe the curve, often making calculations and graphing more manageable.

For an ellipse, parametric equations can generally be written as:

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

where \(t\) is the parameter ranging from 0 to \(2\pi\).

In our example, we employ the equations:

• \(x = -2 + 30 \cos t\)
• \(y = -3 + 20 \sin t\)

This representation is helpful for easily generating points on the ellipse and understanding how they relate to the standard Euclidean form.

Graphing Conics

Graphing conics involves drawing figures such as ellipses, parabolas, and hyperbolas based on their algebraic equations. Conics are curves obtained by intersecting a plane with a double-napped cone.

In the case of an ellipse, knowing its parametric equations helps to plot the curve on a graph, offering a visual representation. By assigning different values to the parameter \(t\), one can easily plot:

• The position of various points on the ellipse
• The overall shape and direction of the ellipse

To graph the given ellipse:

• Use the parametric equations \(x = -2 + 30 \cos t\) and \(y = -3 + 20 \sin t\)
• Plotting these over \(0 \leq t \leq 2 \pi\) yields all points tracing the ellipse

This geometric form enhances comprehension of conic sections and assists in visualizing their algebraic equations.