Problem 38 - College Algebra

Question

Graph each ellipse and give the location of its foci. $$\frac{(x-1)^{2}}{16}+\frac{(y+2)^{2}}{9}=1$$

Step-by-Step Solution

Answer

The center of the ellipse is at (1, -2), the lengths of the major and minor axes are 8 and 6 respectively, and the foci are located at (1 ± \(\sqrt{7}\), -2). The major axis is oriented along the x-axis.

Step 1: Identify the center, and lengths of the major and minor axis

Looking at the given equation, we see that it is already in standard form. From the equation, we can see that (h, k) = (1, -2), a = 4 and b = 3. Since \(a^{2} > b^{2}\), the major axis is along the x axis and its length is 2a = 8. The length of the minor axis is 2b = 6.

Step 2: Identify the orientation of the ellipse

Since \(a^{2} > b^{2}\), the major axis is along the x-axis.

Step 3: Calculate the distance to the foci from the center

We can calculate the distance to the foci from the center with the formula for c which is \(c = \sqrt{a^2 - b^2}\). Substituting the appropriate values, we have \(c = \sqrt{4^2 - 3^2} = \sqrt{7}\). So the foci are at a distance of \(\sqrt{7}\) along the major axis from the center.

Step 4: Graph the ellipse

Start at the center (1, -2). Draw the major axis along the x-axis with a length of 2a = 8 and the minor axis along the y-axis with a length of 2b = 6. Place the foci at a distance of \(\sqrt{7}\) along the major axis from the center. This gives the graph of the ellipse.

Step 5: Identify the location of the foci

The foci are located at a distance of \(\sqrt{7}\) from the center along the major (x) axis. Since the center of the ellipse is at (1, -2), the foci are located at (1 ± \(\sqrt{7}\), -2).