Problem 47 - College Algebra

Question

Graph each ellipse and give the location of its foci. $$\frac{(x-1)^{2}}{2}+\frac{(y+3)^{2}}{5}=1$$

Step-by-Step Solution

Answer

The foci of the ellipse are located at \((1+\sqrt{3}, -3)\) and \((1-\sqrt{3}, -3)\).

Step 1: Identify the Center

The equation for this ellipse is in the format \(\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1\), where (h, k) is the center of the ellipse. By comparing this with the given equation, we see that h=1 and k=-3. So, the center of the ellipse is at (1,-3).

Step 2: Find the Lengths of the Axes

In the same equation, \(a^2\) and \(b^2\) are respectively the squares of the lengths of semi-major and semi-minor axes of the ellipse. The square root of the denominator under x is the semi-major axis length. The square root of the denominator under y is the semi-minor axis length. From the given equation, \(a^2 = 2\), and \(b^2 = 5\). Therefore, \(a = \sqrt{2}\) and \(b=\sqrt{5}\). The lengths of the major and minor axes are twice these values, so the major axis length is \(2\sqrt{5}\) and the minor axis length is \(2\sqrt{2}\).

Step 3: Identify the Foci

The foci are located at a distance of \(c = \sqrt{b^2 - a^2}\) from the center along the major axis. Plugging in our known values, \(c = \sqrt{5 - 2} = \sqrt{3}\). Since the major axis of this ellipse is along the x-axis, the foci are at \((1+\sqrt{3}, -3)\) and \((1-\sqrt{3}, -3)\).

Step 4: Sketch the Ellipse

To sketch the ellipse, first plot the center. Then, draw the major and minor axes, using the lengths calculated in Step 2. Next, plot the foci. Finally, sketch the ellipse shape around these components.