Problem 12 - College Algebra

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. $$ \frac{x^{2}}{100}+\frac{y^{2}}{64}=1 $$

Step-by-Step Solution

Answer

Ellipse: center (0,0), major axis endpoints (±10,0), minor (0,±8), foci (±6,0).

Step 1: Identify the form of the equation

The given equation \( \frac{x^{2}}{100} + \frac{y^{2}}{64} = 1 \) is already in the standard form of an ellipse \( \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 \).

Step 2: Determine the orientation of the ellipse

Since \( a^{2} = 100 \) and \( b^{2} = 64 \), we find that \( a = 10 \) and \( b = 8 \). Since \( a > b \), the major axis is along the x-axis, indicating a horizontal ellipse.

Step 3: Locate the endpoints of the major and minor axes

The center of the ellipse is at the origin \((0, 0)\). The endpoints of the major axis are \(( \, \pm 10, \, 0 \, )\), and the endpoints of the minor axis are \(( \, 0, \, \pm 8 \, )\).

Step 4: Solve for the foci

The foci are located at \(( \, \pm c, \, 0 \, )\), where \( c = \sqrt{a^2 - b^2} \). Calculate \( c = \sqrt{100 - 64} = \sqrt{36} = 6 \). Thus, the foci are \(( \, \pm 6, \, 0 \, )\).

Key Concepts

Standard Form of an Ellipse

In the world of geometry, the standard form of an ellipse is a convenient way to describe its equation. It looks like this: \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \). This equation represents an ellipse centered at the origin (0, 0). The numbers \( a \) and \( b \) are crucial. They determine how far the ellipse stretches along the x-axis and y-axis. Essentially, \( a \) and \( b \) are half the lengths of the ellipse's major and minor axes, respectively.
Understanding the standard form makes it easier to analyze different characteristics of the ellipse, like its size and orientation. When given an equation, if the sum equals 1 after dividing each variable by its respective squared value, you know you’re dealing with an ellipse.
When tackling the exercise, we noticed that our equation, \( \frac{x^2}{100} + \frac{y^2}{64} = 1 \), is in this standard form. Here, \( a^2 = 100 \) and \( b^2 = 64 \), making things straightforward from this starting point.

Major and Minor Axes

Every ellipse has a major and a minor axis. These are the boundaries of the ellipse.
The **major axis** is the longest diameter of the ellipse, stretching from one end to the other through the center. In the standard form, it is represented by \( a \). If \( a > b \), the major axis is horizontal, meaning it runs along the x-axis.
The **minor axis**, on the other hand, is the shortest diameter, denoted by \( b \) in the standard form. If \( a > b \), it means the minor axis runs along the y-axis.
In our exercise, we calculated \( a = 10 \) and \( b = 8 \). Since \( a > b \), the major axis is horizontal with endpoints at \( ( \pm 10, 0) \) and the minor axis is vertical with endpoints at \( (0, \pm 8) \).
This tells us about the ellipse's orientation and size. By understanding this, visualizing the ellipse becomes clearer.

Foci of an Ellipse

The foci of an ellipse are two special points located along the major axis. They are not on the ellipse itself but are key in defining its shape. The sum of the distances from any point on the ellipse to the two foci is constant.
To find the foci, we use a formula: \( c = \sqrt{a^2 - b^2} \). Here, \( c \) tells us how far each focus is from the center. For a horizontal ellipse, like in our problem, the foci are at \( ( \pm c, 0) \).
In our example, substituting the values, we get \( a^2 = 100 \) and \( b^2 = 64 \). Therefore, \( c = \sqrt{100 - 64} = \sqrt{36} = 6 \). So, the foci are located at \( ( \pm 6, 0) \).
Identifying the foci helps in understanding the ellipse's geometry and behavior, as they play a crucial role in its overall balance and form.