Problem 49 - College Algebra with Modeling and Visualization

Question

Find an equation of an ellipse that satisfies the given conditions. Vertices \((\pm 3,2)\) and foci \((\pm 2,2)\)

Step-by-Step Solution

Answer

The equation of the ellipse is \(\frac{x^2}{9} + \frac{(y-2)^2}{5} = 1\).

Step 1: Identify the Center of the Ellipse

The center of the ellipse \((h, k)\) can be identified from the midpoints of the vertices, which are given as \( (\pm 3, 2) \). Since the y-coordinate is the same for both vertices, \(k = 2\), and the midpoint of \(x = 3\) and \(x = -3\) is \(h = 0\). Thus, the center is \( (0, 2) \).

Step 2: Determine the Value of 'a'

The distance from the center to each vertex along the x-axis gives us the value of \(a\). Since the vertices are \( (\pm 3, 2) \), the distance is \(3\). Hence, \(a = 3\).

Step 3: Determine the Value of 'c'

The distance from the center to each focus (foci) is the value of \(c\). Given the foci \( (\pm 2, 2) \), the distance is \(2\). Thus, \(c = 2\).

Step 4: Calculate 'b' using the Relationship between a, b, and c

The relationship between \(a\), \(b\), and \(c\) for an ellipse (when the major axis is horizontal) is \(c^2 = a^2 - b^2\). Plugging in the known values: \(2^2 = 3^2 - b^2\) simplifies to \(4 = 9 - b^2\), giving \(b^2 = 5\) so \(b = \sqrt{5}\).

Step 5: Write the Equation of the Ellipse

The standard form of the ellipse with a horizontal major axis is \[\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1\]Substitute \(h = 0\), \(k = 2\), \(a = 3\), and \(b = \sqrt{5}\):\[\frac{x^2}{9} + \frac{(y-2)^2}{5} = 1\]

Key Concepts

Vertices of an Ellipse

Vertices are critical points on an ellipse that define its shape and orientation. For an ellipse, vertices are the points where the ellipse intersects its major axis. In our problem, the vertices are given as \((\pm 3, 2)\). This tells us two important things:

• The major axis is parallel to the x-axis since the vertices have varying x-coordinates but a constant y-coordinate.
• The distance between these vertices along the x-axis is the length of the major axis. Since the coordinates are \((3,2)\) and \((-3,2)\), the total length is 6 units.

This means each vertex is 3 units away from the center of the ellipse, setting the value of \(a\) (the semi-major axis length) to 3. Learning to identify and use vertices helps us understand the ellipse's dimensions and orientation.

Foci of an Ellipse

The foci are two special points inside the ellipse that help define its shape. The sum of the distances from any point on the ellipse to each focus is constant. In our exercise, the foci are \((\pm 2, 2)\), indicating that they lie on the same line as the vertices, parallel to the x-axis. Each focus is located 2 units from the center.

• For an ellipse with a horizontal major axis, the foci are aligned along the x-axis.
• The distance from the center to each focus, known as \(c\), is 2 units.

Understanding the position of the foci enables us to apply the relationship between \(a\), \(b\), and \(c\), which is essential for deriving the equation of the ellipse. The foci's location is pivotal in determining how "stretched" the ellipse is along its major axis.

Center of an Ellipse

The center of an ellipse is the midpoint between its vertices and foci along the major axis. It's denoted as \((h, k)\), signifying the coordinates around which the ellipse is symmetric. In this exercise, because the vertices and foci share a common y-coordinate, namely 2, we can conclude that the center is \((0, 2)\).

• The center serves as a reference point for the entire ellipse, helping determine the position of vertices and foci.
• All calculations related to the ellipse equation, such as substituting \(h\) and \(k\), rely on knowing the center.

This concept is vital as it directly affects how we construct the equation of the ellipse and interpret its characteristics.

Relationship between a, b, and c in an Ellipse

This relationship is a fundamental characteristic that shapes the structure of an ellipse. Here, \(a\) and \(b\) represent the semi-major and semi-minor axes respectively, while \(c\) is the focal distance. The formula \(c^2 = a^2 - b^2\) describes the connection among these parameters when the major axis is horizontal.

• From our exercise, \(a = 3\), \(c = 2\), so we calculate \(b\) using \(4 = 9 - b^2\), giving \(b^2 = 5\) and \(b = \sqrt{5}\).
• This relationship ensures that the ellipse remains in a specific proportion and orientation, maintaining its geometric properties.

Understanding these parameters allows us to form the ellipse's equation accurately, represented as \(\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1\), as shown in the solution. This formula perfectly encapsulates the symmetry and dimensions of the ellipse in relation to its axes.