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 ellipse is centered at (4,-2), with a major axis length of 5 and a minor axis length of 3. The foci are located at (4,2) and (4,-6).

Step 1: Identify the Center of the Ellipse

The center of the ellipse can be obtained from the given equation. The center is given by (h, k), where \(h = 4\) and \(k = -2\). Thus, the center of the ellipse is (4, -2).

Step 2: Identify the Major and Minor Axes

The lengths of the major and minor axes are given by square roots of the denominators in the equation. The major axis is the bigger one, and the minor axis is the smaller one. From the equation, the major axis \(a = 5\) and minor axis \(b = 3\). Plot the major axis vertically from the center up and down 5 units, and plot the minor axis horizontally from the center to the left and right 3 units.

Step 3: Graph the Ellipse

Using the center and the major and minor axes, draw the ellipse. Begin at the center and draw a vertical line 5 units up and down, and a horizontal line 3 units to the left and right. Then smooth the corners to draw the oval shaped ellipse.

Step 4: Find the Foci

Locating the foci involves using the formula for an ellipse \(c = \sqrt{a^2 - b^2}\), where \(a\) is the length of the major axis, \(b\) is the length of the minor axis and \(c\) is the distance from the center to a foci. Calculating this gives \(c = \sqrt{5^2 - 3^2} = 4\). The foci are located along the major axis, both above and below the center by 4 units. So the foci are (4, -2+4) and (4, -2-4), simplifying to (4,2) and (4,-6).