Question

Use the results of Exercises \(31-34\) to find a set of parametric equations for the line or conic. $$ \text { Ellipse: vertices: }(\pm 5,0) ; \text { foci: }(\pm 4,0) $$

Short answer

The parametric equations for the ellipse are \(x = 5\cos(\theta)\) and \(y = 3\sin(\theta)\), where \(0 \leq \theta \leq 2\pi\).

Step-by-step solution

Identify center, a and b

First, note that the center of the ellipse is at the origin, since the foci and vertices are symmetric with respect to the origin: (0,0). \[a\] is half the distance of the major axis, represented by the distance between the vertices, hence \(a = 5\). \[b\] is found by the relationship \(a^2 = b^2 + c^2\), where \(c\) is the distance from the center to either focus, here \(c = 4\). Solving for \[b\] gives \(b = 3\).

Create parametric equations

Knowing the values of a and b allows writing the parametric equations for the ellipse. For an ellipse centered at the origin, the parametric equations are \(x = a \cos(\theta)\) and \(y = b \sin(\theta)\), where \(0 \leq \theta \leq 2\pi\). Substituting the values of a and b gives the parametric equations as \(x = 5\cos(\theta)\) and \(y = 3\sin(\theta)\).

Key concepts

Parametric Equations

Parametric equations are a way to represent a mathematical relationship by using one or more parameters. Unlike the standard y = f(x) equation that you might be used to, parametric equations define both x and y with respect to an independent parameter, often denoted as \ or \(t\).

For example, consider a point moving along a path on a plane. Rather than describing the path as a function of x alone, we can use parametric equations to describe the x and y coordinates over time or another parameter. This is especially useful for describing curves, like the path of a projectile, the orbit of a planet, or the shape of an ellipse.

In the context of conic sections, such as ellipses, parametric equations allow for a smooth, continuous tracing of the shape. Assuming the center of our conic section is the origin (0,0), the parametric equations for an ellipse generally take the form:\[x = a \cdot cos(\theta)\]\[y = b \cdot sin(\theta)\]where \(a\) and \(b\) are known as the semi-major and semi-minor axes, respectively, and \(\theta\) varies typically from 0 to \(2\pi\), encompassing a full rotation around the ellipse.

Ellipse Standard Form

The standard form of an ellipse equation provides a clear way to identify an ellipse's key features: its center, major axis, minor axis, and foci. When an ellipse is aligned with the axes, the standard form of its equation is:\[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\]

Here, \(a\) and \(b\) are the lengths of the semi-major and semi-minor axes, respectively. The major axis is the longest diameter of the ellipse, and the minor axis is the shortest. The vertices lie at the ends of the major axis, and in the given problem, the vertices are at (±5,0), indicating that \(a = 5\).

To find \(b\), the distance from the center to the foci, often represented by \(c\), is used in the relation \(a^2 = b^2 + c^2\). This equation is derived from the fact that an ellipse is a set of points for which the sum of distances to the two foci is constant. Knowing that in our example the foci are at (±4,0), we conclude that \(c = 4\) and we can solve for \(b\) to identify the shape of the ellipse.

Conic Sections

Conic sections are the curves obtained by intersecting a cone with a plane. They include circles, ellipses, parabolas, and hyperbolas. By adjusting the angle of the intersecting plane relative to the cone's surface, one can obtain the various shapes.

An ellipse is a conic section that appears when the cutting plane intersects the cone at an angle to its base and does not pass through the apex. The uniformity of the shape, where all points on the ellipse are equidistant from the total distance to the foci, gives it unique mathematical properties.

From the perspective of parametric equations, each conic section can be described succinctly and elegantly, with ellipses presenting a unique symmetry that translates into their parametric form. The exercise provided is a perfect illustration of how an ellipse's parametric equations are derived, encapsulating these conic properties in a simple, yet powerful mathematical representation.