Problem 22 - Algebra 2

Question

Find the foci for each equation of an ellipse. Then graph the ellipse. $$ \frac{x^{2}}{64}+\frac{y^{2}}{100}=1 $$

Step-by-Step Solution

Answer

The foci of the given ellipse are located at (0, 6) and (0, -6).

Step 1: Identify a, b and c values

First identify the values of \(a^2\) and \(b^2\) from the given equation, which are 64 and 100 respectively. So, \(a=8\) and \(b=10\). The longer one is always \(b\). Also, as \(b^2 = a^2 + c^2\), we can solve for \(c\) which gives us \(c = \sqrt{b^2 - a^2} = \sqrt{100 - 64} = \sqrt{36} = 6\). This \(6\) is the distance from the center of the ellipse to each of the foci that are located along the major axis.

Step 2: Find the coordinates for the foci

As the major axis is along the y-axis and center of the ellipse is at the origin (0,0), the foci will lie at coordinates (0, \(c\)) and (0, \(-c\)). So the coordinates of the foci are (0, 6) and (0, -6).

Step 3: Graph the ellipse

In the x-y plane, since the origin (0, 0) is the center of the ellipse, draw an ellipse with a horizontal radius of \(a=8\) units and a vertical radius of \(b=10\) units. Also mark the points (0, 6) and (0, -6) as the foci of the ellipse.

Key Concepts

Foci of an Ellipse

In the context of an ellipse, the foci are two special points located on the major axis. The sum of the distances from any point on the ellipse to the two foci is constant. This property helps define the shape of the ellipse.
For the equation \(\frac{x^2}{64} + \frac{y^2}{100} = 1\), you have \(a = 8\) and \(b = 10\). You can calculate the distance to the foci using:

• \(c = \sqrt{b^2 - a^2} = \sqrt{100 - 64} = 6\)

The foci are located at \((0, 6)\) and \((0, -6)\). Placing the foci along the y-axis indicates that the major axis is vertical, and these points are symmetrical relative to the center.

Major Axis of an Ellipse

The major axis of an ellipse is the longest diameter that runs through the center, touching both ends of the ellipse. It is the axis with the greater value between \(a\) and \(b\).
In our example, \(b = 10\) is larger than \(a = 8\), which means the major axis is vertical. This major axis influences the ellipse's shape and size. The total length of the major axis is \(2b = 20\) units.
The ellipse stretches longer along the major axis, and you will find the foci on this same line, ensuring the ellipse's unique form is created.

Graphing Ellipses

Graphing an ellipse involves plotting its center, axes, and sometimes its foci. Start by identifying the axes' lengths, then locate the center. For the equation \(\frac{x^2}{64} + \frac{y^2}{100} = 1\), the center is at the origin (0, 0).
Draw the y-axis length of 10 units and the x-axis length of 8 units radiating from the center:

• Horizontal radius: 8 units
• Vertical radius: 10 units

Next, sketch a smooth curve connecting these radii to form an elongated oval shape. Mark the foci at \((0, 6)\) and \((0, -6)\). All points on the ellipse ensure the sum of the distances to these foci remain equal.

Center of the Ellipse

The center of an ellipse is a central point equally distant from all sides of the ellipse's outline. In standard position, this center is often at the origin (0, 0). This is where both axes (major and minor) intersect.
In the equation \(\frac{x^2}{64} + \frac{y^2}{100} = 1\), the center is clearly at (0, 0).
This point serves as the midpoint for calculating the axes lengths and is crucial for symmetry. An accurate center location aids in neatly drawing the ellipse and ensuring the foci's correct alignment.