Problem 48 - 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 ellipse is centered at (-1, 3) with semi-major axis length \( \sqrt{5} \) and semi-minor axis length \( \sqrt{2} \). The coordinates of the foci are \((-1, 3+ \sqrt{3}) \) and \((-1, 3 - \sqrt{3}) \).

Step 1: Identify the Center

To find the center of the ellipse, we equate the expression inside the parentheses to zero. Hence the center of the ellipse is at \((-1, 3)\).

Step 2: Identify the semi-major and semi-minor axes

The values under x and y are the squares of the semi-major axis and semi-minor axis. The larger denominator is under y, so we have semi-major axis a as \( \sqrt{5} \) and semi-minor axis b as \( \sqrt{2} \).

Step 3: Calculate the Foci

Now we calculate the foci. The distance from the center to each focus is given by \(c = \sqrt{a^2 - b^2}\). Plugging in the values we have, \(c = \sqrt{5-2} = \sqrt{3}\). The foci are \(c\) units above and below the center along the major axis (vertical axis) since we have a > b. Therefore, the foci are at \((-1, 3+ \sqrt{3}) \) and \((-1, 3 - \sqrt{3})\)