Problem 38 - Calculus Early Transcendentals: Pearson New International Edition

Question

Ellipse with foci at \((2,0)\) and \((2,12)\) and a vertex at \((2,14)\)

Step-by-Step Solution

Answer

The equation is \(\frac{(x-2)^2}{28} + \frac{(y-6)^2}{64} = 1\).

Step 1: Identifying the Center

Since the foci are at (2,0) and (2,12) , and a vertex at (2,14), the ellipse is vertically oriented. The center of the ellipse is midway between the foci, which can be found using the midpoint formula on the y-coordinates. The x-coordinate is given by the foci and vertex and remains 2. Thus, the center is (2,6) .

Step 2: Finding the Distance between Center and Vertex (a)

The distance from the center (2,6) to the vertex (2,14) is found by subtracting 6 from 14 to get 8. Therefore, a = 8 .

Step 3: Calculating the Distance between Center and Focus (c)

The distance from the center (2,6) to one of the foci (2,12) is found by subtracting 6 from 12 to get 6. Therefore, c = 6 .

Step 4: Using the Ellipse Equation Formula

For an ellipse with a vertical major axis, the equation is \(rac{(x-h)^2}{b^2} + rac{(y-k)^2}{a^2} = 1\). We know a = 8 and c = 6, so we use the relationship a^2 = b^2 + c^2 to find b^2. Calculating, 8^2 = b^2 + 6^2, results in 64 = b^2 + 36 . Thus, b^2 = 64 - 36 = 28.

Step 5: Writing the Equation of the Ellipse

Given h = 2 and k = 6 and previously calculated, a^2 = 64 and b^2 = 28, the equation is \(rac{(x-2)^2}{28} + rac{(y-6)^2}{64} = 1\).

Key Concepts

Ellipse Equation

An ellipse is a geometric shape resembling an oval. Its equation takes a standard form, allowing us to delve deeply into its features like the center, axes, and foci. The equation can differ depending on the orientation of the ellipse. For a vertically oriented ellipse, it reads:

• \( \frac{(x-h)^2}{b^2} + \frac{(y-k)^2}{a^2} = 1 \)

Here, \((h, k)\) is the center of the ellipse, while \(a\) and \(b\) represent the lengths of the semi-major and semi-minor axes, respectively. Since \(a\) is associated with the larger axis, for a vertical ellipse, \(a\) corresponds to the y-term. Once you know the values of \(a^2\) and \(b^2\), you can effectively describe an ellipse's shape in the coordinate plane.
Calculating these values requires knowing the distances involved with the major axis and utilizing the relationship \(a^2 = b^2 + c^2\), where \(c\) is the distance from the center to each focus.

Major Axis

The major axis of an ellipse is its longest diameter, passing through both foci and the center. In a vertically oriented ellipse, the major axis is parallel to the y-axis.

• This axis spans from one vertex, through the center, to the other vertex.
• The length of the major axis is equal to \(2a\).

In our example, the major axis stretches from \((2, 0)\) to \((2, 14)\) as we identified the vertices at these points. This made the center of the ellipse \((2, 6)\), right between these points on the vertical axis.
Understanding the major axis is vital as it helps verify important parts of the ellipse equation. It primarily affects the \((y-k)^2\) portion of the equation since \(a\) measures how the ellipse stretches vertically.

Foci of Ellipse

Foci (singular: focus) are two special points on an ellipse from which the sum of the distances to any point on the ellipse is constant. This constant sum always equals the length of the major axis.

• In our case, the foci are at \((2, 0)\) and \((2, 12)\).
• Their placement along the minor axis signifies how spread out the shape is.

The distance between the center and each focus is \(c\), and we use this value in the equation \(a^2 = b^2 + c^2\). Identifying these points is crucial because they are key to establishing the focal properties of an ellipse, directly influencing the equation's parameters by helping us calculate \(b^2\) when \(a^2\) and \(c\) are known. Their positions validate the unique geometric nature of the ellipse.

Center of Ellipse

The center of an ellipse is the midpoint between the two foci. This central point is crucial for positioning the ellipse in the coordinate plane. Using the midpoint formula, midpoints are averaged to find the center location.

• For vertical ellipses, you specifically consider the y-coordinates of the foci.
• In our setup, the foci are \((2, 0)\) and \((2, 12)\), so the center is \((2, 6)\).

The x-coordinate remains constant as \(2\) since it is consistent across the positions of foci and vertex on the vertical axis. Establishing the center is foundational because it sets a reference point for the major and minor axes' scope. It allows access to other calculations by establishing a clear geometric base for further exploration of the ellipse's properties.