Problem 23 - College Algebra with Modeling and Visualization

Question

Find an equation of the ellipse, centered at the origin, satisfying the conditions. Foci \((\pm 5,0),\) vertices \((\pm 6,0)\)

Step-by-Step Solution

Answer

The equation of the ellipse is \(\frac{x^2}{36} + \frac{y^2}{11} = 1\).

Step 1: Identify Key Features of the Ellipse

The ellipse is centered at the origin, so its general equation will be \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\). Since the foci are along the x-axis (\((\pm 5,0))\)), the major axis is also along the x-axis. The vertices \((\pm 6,0)\) indicate that \(a = 6\).

Step 2: Relate Distance to Foci and Vertices

The distance from the center to each focus (\(c\)) is 5, so \(c = 5\). The distance from the center to each vertex (\(a\)) is 6, so \(a = 6\). This gives us \(c^2 = a^2 - b^2\).

Step 3: Calculate the Semi-minor Axis

Using the relationship \(c^2 = a^2 - b^2\), plug in the known values: \(5^2 = 6^2 - b^2\) or \(25 = 36 - b^2\). Solving gives us \(b^2 = 11\).

Step 4: Write the Equation of the Ellipse

Insert the values of \(a^2 = 36\) and \(b^2 = 11\) into the standard equation format, yielding \(\frac{x^2}{36} + \frac{y^2}{11} = 1\).

Key Concepts

Foci of an Ellipse

An ellipse has two special points known as the foci (plural of focus). Understanding their role is crucial in grasping how ellipses work. In the context of ellipses, the foci are two fixed points, and the sum of the distances from these points to any point on the ellipse is constant. This property ensures that the shape forms an elongated circle that varies in width or height across its major and minor axes. For our particular example, the foci are located at

• positions (\(+5,0\)) and (\(-5,0\)) on the x-axis, reflecting that they lie exactly at equal distances from the origin.

The mathematical significance of the foci is realized through the calculation of the value called \(c\), which represents the distance from the center of the ellipse to each of the foci. In the equation \(c^2 = a^2 - b^2\), this value helps define the relationship between the ellipse's axes.

Ellipses have their unique beauty and properties, and recognizing the foci is essential in describing these shapes mathematically.

Vertices of an Ellipse

The vertices of an ellipse are the endpoints of its major axis, the longest diameter that runs through the center of the shape. The vertices give us vital information about an ellipse’s size and orientation. In the case of a standard ellipse that is centered at the origin, such as the one we are examining, the vertices lie along the major axis. This axis can either be on the x-axis or y-axis, depending on the alignment of the ellipse.

For our given example, the vertices are located at

• positions (\(+6,0\)) and (\(-6,0\)) on the x-axis.

These points signify that the ellipse extends outwards horizontally over a length of 12 units, with each vertex being 6 units away from the center. The horizontal placement of the vertices also confirms that the major axis, and indeed the longer portion of the ellipse, span along the x-axis.

By determining the vertices, we can directly derive the value of \(a\), which is the semi-major axis, showing how far the ellipse stretches from its own center to its vertex.

Semi-minor Axis of an Ellipse

The semi-minor axis is one of the fundamental dimensions of an ellipse, lesser in terms of extent than the semi-major axis. In any ellipse, it is always perpendicular to the semi-major axis and provides the shortest path through the center from one side of the ellipse to the other.

In the example we've been working with, since we've identified that the major axis is the x-axis, the semi-minor axis naturally runs along the y-axis.

Using the equation \(c^2 = a^2 - b^2\), where we've already calculated

• \(c = 5\) and \(a = 6\)

we find that \(b^2\) equals 11, and hence \(b\), the length of the semi-minor axis, is the square root of 11.

This calculation is pivotal because it helps finalize the full description of the ellipse. Knowing both the semi-major and semi-minor axes allows for complete visualization and mathematical representation, ensuring an exact understanding of the ellipse’s proportions and dimensions.