Problem 47 - College Algebra with Modeling and Visualization

Question

Find an equation of an ellipse that satisfies the given conditions. Center \((2,1),\) focus \((2,3),\) and vertex \((2,4)\)

Step-by-Step Solution

Answer

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

Step 1: Determine the orientation of the ellipse

The center of the ellipse is at (2,1), and the focus and vertex both have the same x-coordinate (2) as the center. This indicates that the ellipse is vertically oriented.

Step 2: Identify the values of a, b, and c

The distance from the center to the vertex along the major axis is the length of the semi-major axis, denoted as \(a\). So, \(a = |4 - 1| = 3\). The distance from the center to the focus is \(c = |3 - 1| = 2\).

Step 3: Use the formula \(c^2 = a^2 - b^2\) to find b

Substitute the known values: \(c^2 = 4\) and \(a^2 = 9\). Using the relation \(c^2 = a^2 - b^2\), we get \(4 = 9 - b^2\). Solving for \(b^2\) gives \(b^2 = 5\).

Step 4: Write the equation of the ellipse in standard form

For a vertically oriented ellipse centered at \((h,k) = (2,1)\), the standard form of the equation is \(\frac{(x-h)^2}{b^2} + \frac{(y-k)^2}{a^2} = 1\). Substituting \(b^2 = 5\) and \(a^2 = 9\), the equation becomes: \(\frac{(x-2)^2}{5} + \frac{(y-1)^2}{9} = 1\).

Key Concepts

Center of an Ellipse

The center of an ellipse is one of its most important features. It serves as a reference point for the overall position of the ellipse in a coordinate system. In our exercise, the center is given as \((2, 1)\). This means:

• The horizontal position is at \(x = 2\)
• The vertical position is at \(y = 1\)

The center acts as the midpoint between the vertices along the major axis and between the foci of the ellipse. Because ellipses are symmetrical, the center divides the major and minor axes into equal halves. When an ellipse is oriented vertically (as in this exercise), its equation considers the coordinates \((h, k)\), where \(h\) and \(k\) are the x and y coordinates of the center, respectively. Placing the center at \( (h,k) \) simplifies writing the standard form of the ellipse's equation.

Focus and Vertex

Understanding the focus and vertex is critical when dealing with ellipses. They define the shape and orientation of the ellipse.*Focus:* This point lies inside the ellipse and helps to determine its width and how elongated it is. In the given exercise, the focus is at \((2, 3)\), aligning vertically with the center. The distance from the center to the focus is denoted by \(c\). Calculating it using the formula:

• Distance \(c = |3 - 1| = 2\)

*Vertex:* It is located at the terminus of the major axis and lies on the ellipse. The given vertex is at \((2, 4)\), also vertically aligned. The distance from the center to the vertex determines the length of the semi-major axis:

• Distance \(a = |4 - 1| = 3\)

Standard Form of Ellipse

The standard form of the ellipse's equation helps define its size and shape based on orientation. For an ellipse centered at \((h, k)\), the equation differs depending on whether the major axis is horizontal or vertical. When the major axis is vertical, as in this exercise, the standard form of the equation is:\[\frac{(x-h)^2}{b^2} + \frac{(y-k)^2}{a^2} = 1\]Here, \(h\) and \(k\) come from the center, \((2, 1)\). The variables \(a\) and \(b\) are the lengths of the semi-major and semi-minor axes, respectively:

• \(a = 3\), representing the semi-major axis
• \(b^2 = 5\), calculated using the relation \(c^2 = a^2 - b^2\)

Incorporating these into the standard form, the equation of this specific ellipse is:\[\frac{(x-2)^2}{5} + \frac{(y-1)^2}{9} = 1\]This equation fully represents the ellipse with the center \((2, 1)\), a focus at \((2, 3)\), and a vertex at \((2, 4)\).