Problem 37 - College Algebra

Question

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

Step-by-Step Solution

Answer

The center of the ellipse is at (2, 1), and it has a semi-major axis of 3 along the x-axis and a semi-minor axis of 2 along the y-axis. The foci of the ellipse are located at \((2 - \sqrt{5}, 1)\) and \((2 + \sqrt{5}, 1)\).

Step 1: Identify the center, semi-major axis and semi-minor axis

The standard form of an ellipse equation gives us the center, semi-major axis and semi-minor axis directly. In this case, the center is at (h, k) = (2, 1). The semi-major axis \(a\) is the square root of the larger denominator and semi-minor axis \(b\) is the square root of the smaller denominator, so \(a = \sqrt{9} = 3\) and \(b = \sqrt{4} = 2\). The semi-major axis is along the x-axis and the semi-minor axis is along the y-axis.

Step 2: Find the location of the foci

The foci are located along the major axis, a distance of \(\sqrt{a^2 - b^2}\) from the center. This gives us \(\sqrt{9 - 4} = \sqrt{5}\). Therefore, the foci are located at \((2 - \sqrt{5}, 1)\) and \((2 + \sqrt{5}, 1)\).

Step 3: Sketch the ellipse

Mark the center of the ellipse at (2, 1) on a graph. From there, draw the semi-major axis along the x-axis 3 units in both directions, and the semi-minor axis along the y-axis 2 units in both directions. Sketch the ellipse that touches the ends of these axes. Mark the foci inside the ellipse.