Problem 39 - Calculus Early Transcendentals

Question

Find an equation for the conic that satisfies the given conditions. Ellipse, foci\((0,2),(0,6), \quad\) vertices \((0,0),(0,8)\)

Step-by-Step Solution

Answer

The equation is \(\frac{x^2}{12} + \frac{(y-4)^2}{16} = 1.\)

Step 1: Determine the Orientation of the Ellipse

Given foci (0,2) and (0,6), we notice that the ellipse is vertical as the foci have the same x-coordinate. The vertices at (0,0) and (0,8) further confirm this vertical orientation. This implies that the transverse axis is along the y-axis.

Step 2: Identify the Center of the Ellipse

The center of the ellipse is midway between the vertices. Therefore, calculate the midpoint of vertices (0,0) and (0,8). The center, (h,k), is given by \(h = 0, \) and \(k = \frac{0+8}{2} = 4. \) So the center is (0,4).

Step 3: Determine the Values for a and b

From the vertices (0,0) and (0,8), the distance from the center to a vertex is \(a = 4.\) The foci show us that the distance from the center to either focus is \(c = 2.\) The relationship \(c^2 = a^2 - b^2\) allows us to find b. \[2^2 = 4^2 - b^2 \] \[4 = 16 - b^2 \] \[b^2 = 12 \] Therefore, \(b = \sqrt{12}.\)

Step 4: Write the Equation of the Ellipse

For a vertical ellipse centered at (h, k), the equation is: \[\frac{(x-h)^2}{b^2} + \frac{(y-k)^2}{a^2} = 1\] Substituting the known values \((h, k) = (0, 4), \ b^2 = 12, \ a^2 = 16,\) the equation becomes: \[\frac{x^2}{12} + \frac{(y-4)^2}{16} = 1.\]

Key Concepts

Ellipse Equation

An ellipse is a type of conic section that can be represented with a standard equation, which depends on its orientation.
To write the equation for an ellipse, we need to identify whether the ellipse is oriented horizontally or vertically.

• For a horizontal orientation, the equation is \(\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1\).
• For a vertical orientation, the equation takes the form \(\frac{(x-h)^2}{b^2} + \frac{(y-k)^2}{a^2} = 1\).

In our example, as the foci and vertices align vertically along the y-axis, the ellipse has a vertical orientation.
The center of this ellipse is at \(h = 0, k = 4\). The values identified for \(a\) and \(b^2\) are crucial for the equation. Specifically, \(a = 4\) and \(b^2 = 12\). With these values, the equation becomes easy to formulate: \(\frac{x^2}{12} + \frac{(y-4)^2}{16} = 1\). This equation uniquely describes the ellipse with the given orientation and characteristics.

Conic Properties

Conic sections, like ellipses, are shapes formed by the intersection of a plane and a double-napped cone.
Each type of conic section – circles, ellipses, parabolas, and hyperbolas – has distinct properties.

• An ellipse in particular is defined by its two axes: the major and minor axes.
• The major axis is the longer one and in the problem is aligned vertically. The length is determined by \(2a\).
• The minor axis is shorter, stretching laterally according to \(2b\).

In the case of our ellipse, the distance from the center to each vertex along the y-axis tells us that \(a = 4\).
The relationship \(c^2 = a^2 - b^2\) points out that from \(c = 2, a = 4\), we solve for \(b^2\) as 12.
These relationships guide us to build the equation that showcases the ellipse's symmetry and alignment.

Foci and Vertices

The special points known as foci and vertices significantly define an ellipse, affecting its shape and size.
Every ellipse possesses two foci, residing along its major axis. These foci are not on the ellipse, but inside its boundary.

• For a vertically oriented ellipse like in our example, the foci are at \(0, 2\) and \(0, 6\).
• Vertices represent the intersection of the ellipse with its major axis.
• For this ellipse, the vertices are located at \(0, 0\) and \(0, 8\).

The distance between a vertex and the center is represented by \(a\), whereas \(c\) is the distance from the center to each focus.
The relationship \(c^2 = a^2 - b^2\) governs how far the foci are positioned from the center.
With the vertices and foci identified, constructing the equation becomes straightforward, highlighting these fundamental properties of ellipses.