93–94. Parametric equations of ellipses Find parametric equations (not unique) of the following ellipses (see Exercises 91–92). 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.)

Verified step by step guidance

1. Identify the center of the ellipse as \((-2, -3)\), the lengths of the major and minor axes as 30 and 20, respectively, and note that the axes are parallel to the x- and y-axes.
2. Recall the standard parametric equations for an ellipse centered at the origin with semi-major axis \(a\) and semi-minor axis \(b\): \[x = a \cos(t), \quad y = b \sin(t)\] where \(t\) is the parameter varying from \(0\) to \(2\pi\).
3. Calculate the semi-major axis \(a\) and semi-minor axis \(b\) by dividing the lengths of the axes by 2: \[a = \frac{30}{2} = 15, \quad b = \frac{20}{2} = 10\]
4. Shift the parametric equations to account for the center \((-2, -3)\) by adding the center coordinates to the \(x\) and \(y\) components: \[x = -2 + 15 \cos(t), \quad y = -3 + 10 \sin(t)\]
5. Write the equation of the ellipse in terms of \(x\) and \(y\) by using the standard form centered at \((-2, -3)\): \[\frac{(x + 2)^2}{15^2} + \frac{(y + 3)^2}{10^2} = 1\]

Key Concepts

Parametric Equations of an Ellipse

Parametric equations express the coordinates of points on an ellipse as functions of a parameter, usually t. For an ellipse centered at the origin with axes aligned to the coordinate axes, the standard form is x = a cos(t), y = b sin(t), where a and b are the semi-major and semi-minor axes. These equations trace the ellipse as t varies from 0 to 2π.

Shifting the Center of an Ellipse

When an ellipse is not centered at the origin, its parametric equations must be adjusted by adding the center coordinates to the standard form. If the center is at (h, k), the equations become x = h + a cos(t) and y = k + b sin(t). This shift translates the ellipse without changing its shape or orientation.

Relationship Between Parametric and Cartesian Forms

The Cartesian equation of an ellipse relates x and y directly, typically in the form ((x - h)^2)/a^2 + ((y - k)^2)/b^2 = 1. Parametric equations provide a way to generate points on the ellipse, while the Cartesian form describes all points satisfying the ellipse's geometric definition. Understanding both forms helps in graphing and analyzing ellipses.