In Exercises 45–52, 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

Verified step by step guidance

1. Identify the center of the ellipse as \((-2, 3)\).
2. Determine the distance from the center to the vertices along the major axis, which is 5 units, indicating the semi-major axis \(a = 5\).
3. Determine the distance from the center to the endpoints of the minor axis, which is 2 units, indicating the semi-minor axis \(b = 2\).
4. Write the standard form of the parametric equations for an ellipse centered at \((h, k)\) with semi-major axis \(a\) and semi-minor axis \(b\): \(x = h + a \cos(t)\) and \(y = k + b \sin(t)\).
5. Substitute \(h = -2\), \(k = 3\), \(a = 5\), and \(b = 2\) into the parametric equations to get \(x = -2 + 5 \cos(t)\) and \(y = 3 + 2 \sin(t)\).

Key Concepts

Parametric Equations

Parametric equations express the coordinates of points on a curve as functions of a variable, typically time (t). For conic sections like ellipses, these equations allow us to describe the shape and position of the curve in a more flexible way than standard Cartesian equations. By defining x and y in terms of t, we can easily manipulate and analyze the geometric properties of the conic.

Ellipse Properties

An ellipse is defined by its center, major axis, and minor axis. The distance from the center to the vertices along the major axis determines the ellipse's width, while the distance to the endpoints of the minor axis determines its height. Understanding these properties is crucial for constructing the parametric equations, as they dictate the lengths and orientations of the axes.

Coordinate System

The coordinate system provides a framework for locating points in a plane using ordered pairs (x, y). In the context of the ellipse described, the center at (-2, 3) serves as the reference point for determining the positions of the vertices and endpoints of the minor axis. Familiarity with how to translate these points into parametric equations is essential for accurately representing the ellipse.