Problem 41 - College Algebra

Question

Graph each ellipse and give the location of its foci. $$\frac{(x-4)^{2}}{9}+\frac{(y+2)^{2}}{25}=1$$

Step-by-Step Solution

Answer

The center of the ellipse is at (4, -2), the semi-major axis has a length of 5 units and the semi-minor axis has a length of 3 units. The foci are located at (0, -2) and (8, -2).

Step 1: Identify the Center

The center of the ellipse can be found in the brackets of the equation. The x-coordinate of the center is h and the y-coordinate of the center is k. The center is given by \( (h, k) = (4, -2) \). This is because in the equation, (x-4) indicates a shift to the right by 4 and (y+2) indicates a shift down by 2.

Step 2: Identify the lengths of the semi-major and semi-minor axis

The values under the x and y terms of the ellipse equation represent \(a^{2}\) and \(b^{2}\) respectively. Therefore, \(a^{2} = 9\) (so, \(a = 3\)) and \(b^{2} = 25\) (so, \(b = 5\)).

Step 3: Find the foci

The foci of an ellipse are located along the major axis, a distance of c units from the center, where \(c = \sqrt{b^{2}-a^{2}}\). Here, \(c = \sqrt{25 - 9} = 4\). Thus, the foci are at the points \((4-4, -2) = (0, -2)\) and \((4+4, -2) = (8, -2)\).

Step 4: Graph the Ellipse

First plot the center of the ellipse at (4, -2). Then draw the major axis from the center extending 5 units vertically in both directions and the minor axis extending 3 units horizontally from the center. Finally, mark the foci, which are 4 units to the left and right of the center on the x-axis.