Problem 47 - Algebra and Trigonometry

Question

Use your answers from Exercises 41–44 and the parametric equations given in Exercises 41–44 to find a set of parametric equations for the conic section or the line. Ellipse: Center: (-2, 3); Vertices: 5 units to the left and right of the center; Endpoints of Minor Axis: 2 units above and below the center

Step-by-Step Solution

Answer

The parametric equations for the described ellipse are: x = -2 + 5*cos(t), y = 3 + 2*sin(t)

Step 1 - Identify Given Information

The center of the ellipse is (-2, 3). The distances to the vertices are 5 units, meaning the horizontal axis length is 5. The distance to the endpoints of the minor axis is 2 units, indicating the vertical axis length is 2.

Step 2 – Construct the Parametric Equations

Use the standard form of the parametric equations and adjust them according to the center and the lengths of the horizontal and vertical axis. For the horizontal axis, the equation will look like: x = -2 + 5*cos(t), which reflects the center position and distance to vertices. Correspondingly, for the vertical axis, the equation will be: y = 3 + 2*sin(t), where 3 is the y coordinate of the center and 2 is the distance to the endpoints of the minor axis.

Step 3 – Present the Final Answer

The parametric equations for an ellipse with given conditions are: x = -2 + 5*cos(t), y = 3 + 2*sin(t)

Key Concepts

Conic Sections

Imagine slicing a cone with a plane at different angles, and you'll uncover the fascinating family of shapes known as conic sections. These shapes include circles, ellipses, hyperbolas, and parabolas. Each one has a distinct set of properties and equations that describe their specific curves.

An ellipse, one of the conic sections, is an oval shape that can be seen in everyday objects like a garden path or an orbit of a planet. It's defined mathematically by its major and minor axes. The major axis runs through the longest part of the ellipse, while the minor axis is the shorter, perpendicular bisector. When you draw an ellipse on a coordinate plane, its center, the major axis, and the minor axis create a roadmap to understanding its geometry and size.

Ellipse Standard Form

The standard form of an ellipse's equation provides a blueprint for its shape and size. Think of it as the DNA of the ellipse, giving all necessary information to draw it precisely on a coordinate plane. The equation is \(\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1\), where \(h,k\) represents the center of the ellipse, and \(a\) and \(b\) denote the lengths of the semi-major and semi-minor axes respectively. If \(a > b\), the ellipse stretches further horizontally; if \(b > a\), it extends more vertically.

Understanding this form helps break down the ellipse's attributes to understandable chunks, like the center point telling you where to place your compass and the \(a\) and \(b\) values guiding how wide you need to draw.

Parametric Representation

Parametric equations offer a dynamic way to describe curves like ellipses. Instead of a single equation, parametric representation breaks down the curve into two equations that define the x and y coordinates separately, based on a third variable, usually denoted as \(t\). For an ellipse, the x and y coordinates can be expressed as \(x = h + a\cos(t)\) and \(y = k + b\sin(t)\), where \(t\) is a parameter that typically represents an angle.

The beauty of parametric equations lies in how they allow us to trace the curve of the ellipse as \(t\) varies, often from \(0\) to \(2\pi\). For instance, when we input different values of \(t\) into the equations, we get corresponding points on the ellipse, and as \(t\) changes, these points form the familiar oval shape. The parametric representation is particularly handy when dealing with rotations or animations as it aligns with the motion principles.