Problem 38 - Precalculus with Limits

Question

In Exercises 33-40, use the results of Exercises 29-32 to find a set of parametric equations for the line or conic. Ellipse: vertices: \((3, 7), (3, -1); \quad\) foci: \((3, 5), (3, 1)\)

Step-by-Step Solution

Answer

The parametric equations for the ellipse are \( x = 3 + 4*cos(t) \) and \( y = 3 + 2*sin(t) \)

Step 1: Determine the center of the ellipse

An ellipse's center is halfway between its vertices and also halfway between its foci. Here, since the x coordinates of the vertices and foci are the same, we know that the x coordinate of the center is 3. The y coordinate is halfway between the y coordinates of the vertices, \(7\) and \(-1\), or \(7+(-1)/2=3\). So, the center of the ellipse is \((3, 3)\).

Step 2: Calculate the lengths of the semi-major and semi-minor axes

The distance from the center to a vertex is the length of the semi-major axis, \(a\), and the distance from the center to a focus is the length of the semi-minor axis, \(b\). So, \(a = |7-3| = 4\) and \(b = |5-3| = 2\).

Step 3: Substitute the values into the ellipse equation

Now substitute the values obtained for the center and the lengths of the axes into the ellipse equation to get the standard form of the equation : \((x-3)^2/4 + (y-3)^2/2 = 1\).

Step 4: Write the parametric equations

Finally, we write the parametric equations for the ellipse. Let \(t\) be the parameter. Then \(x = h + a*cos(t) = 3 + 4 *cos(t) \) and \(y = k + b*sin(t) = 3 + 2*sin(t) \). These are the parametric equations for the given ellipse.

Key Concepts

Ellipse

An ellipse is a smooth, closed curve that is shaped somewhat like an elongated circle. In mathematics, it's defined as the set of all points such that the sum of the distances from two fixed points, called foci, is constant.
To visualize an ellipse, imagine stretching a circle along one axis, turning it into an oval shape.

• The longest diameter of the ellipse, passing through its center and both foci, is called the major axis.
• The shortest diameter, perpendicular to the major axis through the center, is known as the minor axis.

Ellipses have unique properties and are found frequently in physics and engineering, such as in the orbits of planets and the design of reflective surfaces.

Vertices

Vertices of an ellipse are the points where the ellipse intersects its major axis. These points are the farthest from each other on the ellipse and represent its 'major end-points.'
In the provided exercise, the vertices are located at

• (3, 7)
• (3, -1)

These vertices help determine the orientation and the bounds of the ellipse.
For a vertically oriented ellipse like in the exercise, the vertices have the same x-coordinate, which simplifies calculations when determining the center or the axes.

Semi-Major Axis

The semi-major axis is half of the longest diameter of the ellipse. It extends from the center of the ellipse to one vertex along the major axis.
For our understanding:

• It is denoted by the symbol \(a\).
• The length of the semi-major axis determines how 'stretched' the ellipse is.
• For the ellipse in the exercise, the semi-major axis is vertical, with a length \(a = 4\).

The semi-major axis plays a crucial role when writing the standard form or parametric equations of an ellipse.

Semi-Minor Axis

The semi-minor axis of an ellipse is half of its shortest diameter and is designated as \(b\). It stretches from the center to the curve itself along the minor axis.
In mathematical terms:

• The length of the semi-minor axis indicates how 'narrow' the ellipse is compared to its length.
• In our exercise example, \(b = 2\).

Understanding the semi-minor axis helps when graphing the ellipse or determining its parametric equations. This axis is perpendicular to the semi-major axis and plays a critical supportive role in shaping the ellipse.