Problem 5 - College Algebra

Question

In Exercises \(1-18,\) graph each ellipse and locate the foci. $$\frac{x^{2}}{25}+\frac{y^{2}}{64}=1$$

Step-by-Step Solution

Answer

The semi-major and semi-minor axes of the ellipse are 8 and 5 respectively, oriented vertically. The foci are located at (0, ±4.898979).

Step 1: Identify the lengths of the semi-axes

The form of the equation is \(\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1\), from which \(a = \sqrt{25} = 5\) and \(b = \sqrt{64} = 8\).

Step 2: Determine the major and minor semi-axes

Since \(b > a\), the semi-major axis is along the y-axis and the semi-minor axis is along the x-axis.

Step 3: Calculate the foci

To calculate the foci, use the formula \(c = \sqrt{b^2 - a^2}\). Substituting the values of a and b into the equation gives \(c = \sqrt{8^2 - 5^2} = 4.898979 \). The foci are on the major axis, which is on the y-axis in this case, so the foci are at (0, ±4.898979).

Step 4: Sketch the ellipse

Use the determined features to sketch the ellipse. Plot the center of the ellipse at the origin (0,0), plot the vertices at (0,-8) and (0,8), and also plot the foci at (0,-4.898979) and (0,4.898979). Draw the ellipse which will be vertical due to major axis being y-axis.

Key Concepts

Semi-axes

An ellipse is a smooth, rounded shape, where each point's distance from two fixed points (called foci) is constant. The semi-axes are key components of an ellipse, representing half the lengths of its axes. In the standard ellipse equation, which is given as \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), the terms 'a' and 'b' correspond to the semi-axes lengths. Here, 'a' and 'b' take the square root of the denominators of the terms: \(a = \sqrt{25} = 5\) and \(b = \sqrt{64} = 8\). This gives us the lengths of the semi-axes.

When discussing semi-axes:

• 'a' represents the semi-axis along the x-direction, also known as the semi-minor axis if \(a < b\).
• 'b' represents the semi-axis along the y-direction, known as the semi-major axis if \(b > a\).

Understanding the semi-axes is critical as they define the orientation and dimensions of the ellipse.

Foci

The foci (plural of focus) are two special points inside an ellipse. Each point on the ellipse has the same total distance from each focus, embodying the unique geometric property of ellipses. Calculating the positioning of these foci is vital for sketching an accurate ellipse.

To find the distance from the center to each focus, we use the formula \(c = \sqrt{b^2 - a^2}\). For our ellipse:

• Calculate \(c = \sqrt{64 - 25} = \sqrt{39} \approx 4.898979\).

Hence, the foci are located symmetrically about the center along the major axis, which in this case is the y-axis.

Therefore, the foci are positioned at (0, ±4.898979). This means they lie on a vertical line through the origin, showing that the major axis runs vertically.

Major and Minor Axes

The major and minor axes are the two principal measurements of an ellipse. These axes help define its shape and orientation in a coordinate system.

Here's how they work:

• The major axis is the longest diameter that spans across the ellipse and determines which direction the ellipse stretches most. In our given ellipse, with 'b' being larger than 'a', the major axis runs along the y-axis, vertically.
• The minor axis is the shorter diameter and is perpendicular to the major axis. Here, it runs horizontally along the x-axis.

To visualize this:

• The endpoints of the major axis, known as the vertices, are at (0, ±8), marking the topmost and bottommost points of the ellipse.
• The endpoints of the minor axis are at (±5, 0), highlighting the farthest left and right points of the ellipse.

The intersection of these axes is the center of the ellipse, located at the origin (0,0). Recognizing the major and minor axes allows you to accurately sketch the ellipse, ensuring it is properly oriented in the plane.