Problem 18 - College Algebra with Corequisite Support

Question

For the following exercises, write the equation of an ellipse in standard form, and identify the end points of the major and minor axes as well as the foci. \(\frac{(x-7)^{2}}{49}+\frac{(y-7)^{2}}{49}=1\)

Step-by-Step Solution

Answer

The ellipse is a circle with end points (0, 7), (14, 7), (7, 0), and (7, 14). The foci are (7, 7).

Step 1: Identify the Standard Ellipse Form

The given ellipse equation is \(\frac{(x-7)^2}{49} + \frac{(y-7)^2}{49} = 1\). The standard form for an ellipse is \(\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1\), where \((h, k)\) is the center, \(a\) is the semi-major axis length if \(a > b\), and \(b\) is the semi-minor axis length. In this form, \((h, k) = (7, 7)\).

Step 2: Determine the Type of Ellipse

When \(a^2 = 49\) and \(b^2 = 49\), it means \(a = b\). Hence, this ellipse is a special type called a circle, with the same length for both the semi-major and semi-minor axes.

Step 3: Compute Major and Minor Axis End Points

Both axes have equal lengths because \(a = b = 7\). Since the center of the circle is \((7, 7)\), the end points of both axes are \((7 \pm 7, 7) = (0, 7)\) and \((14, 7)\) for the horizontal axis, and \((7, 7 \pm 7) = (7, 0)\) and \((7, 14)\) for the vertical axis.

Step 4: Determine the Foci

For a circle, the foci coincide with the center. Therefore, the foci are at \((7, 7)\).

Key Concepts

Standard Form

In the realm of geometry, particularly when dealing with ellipses, equations often appear in a specific format known as the "standard form." The standard form of an ellipse is expressed as \(\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1\). Here, \((h, k)\) represents the center of the ellipse, where \(h\) and \(k\) are the x and y coordinates of this center, respectively.

The terms \(a\) and \(b\) denote the lengths of the semi-major and semi-minor axes. It's important to compare these values because in a typical ellipse, one will be larger than the other. However, if both are equal, as with our example \(\frac{(x-7)^2}{49} + \frac{(y-7)^2}{49} = 1\), the figure represented is not an ellipse but rather a circle.

Understanding the structure of the standard form helps identify crucial features of the ellipse or circle, such as the lengths of the axes and the position of the center.

Major Axis

The major axis of an ellipse is the longest diameter that runs through the center of the ellipse. It is crucial for determining the shape and orientation of the ellipse in space. In standard form equations, if \(a > b\), then the major axis runs along the x-direction. Conversely, if \(b > a\), it runs along the y-direction.

For our specific case \(\frac{(x-7)^2}{49} + \frac{(y-7)^2}{49} = 1\), we find that \(a = b = 7\). This implies there isn't a predominant major axis, as both axes are equal in length.

Thus, for this example, we are dealing with a circle rather than an ellipse, demonstrating that the concept of a major axis doesn't apply to a well-rounded circle where \(a = b\). However, when present, the major axis end points provide key coordinates that outline the span and orientation of the ellipse.

Minor Axis

Similarly, the minor axis in an ellipse pertains to the shortest diameter that also passes through the center point. Under usual circumstances in ellipses, this differentiates which axis is smaller, providing a crucial description of the ellipse's shape.

Referring back to our example \(\frac{(x-7)^2}{49} + \frac{(y-7)^2}{49} = 1\), the values are \(a = b = 7\), meaning that the axes are equal, signifying that this is not an ellipse per se, but a circle. In the case of a circle, the concept of a minor axis dissolves as the dimensions along both axes are identical.

Nevertheless, understanding the minor axis is critical in scenarios involving actual ellipses, as it aids in determining its symmetry and eccentricity.

Foci

Foci are a defining feature of an ellipse, providing insight into the nature of its shape. By mathematical convention, the foci are two specific points such that the sum of the distances from any point on the ellipse to these foci is constant.

However, in the special case where an ellipse becomes a circle, as seen in our exercise, the concept of separate foci does not apply, and they coalesce with the center itself. If we look back to our circle with the equation \(\frac{(x-7)^2}{49} + \frac{(y-7)^2}{49} = 1\), the center \((7, 7)\) serves as the sole focus for this circle since they are indistinguishable.

In the broader context, comprehending foci helps in understanding ellipses, notions of eccentricity, and how they stretch along the major axis.

Circle

The concept of a circle is fundamentally an ellipse where the semi-major and semi-minor axes are of equal length, making it perfectly symmetrical along all axes. In the equation \(\frac{(x-7)^2}{49} + \frac{(y-7)^2}{49} = 1\), we observe this symmetry as both denominators are identical, indicating \(a = b = 7\).

A circle is characterized by its center and radius, both of which define its complete structure. Here, the center is at \((7, 7)\), and with \(a = b = 7\), the radius is also 7. In geometry, recognizing when an ellipse becomes a circle simplifies working with the equation due to its uniformity.

By grasping these characteristics, solving problems related to ellipses and circles becomes much clearer, allowing students to easily transition between these topics.