Question: Which points are the foci of the ellipse? \[ \begin{array}{c|c} (-5, -2) \text{ and } (-5, 0) &…

Which points are the foci of the ellipse?

\[ \begin{array}{c|c} (-5, -2) \text{ and } (-5, 0) & \\ \hline (-8, -1) \text{ and } (-2, -1) & \\ \hline (-5, -4) \text{ and } (-5, 2) & \\ \hline (-6, -1) \text{ and } (-5, -1) & \end{array} \]

Solution

To find the foci of the ellipse, we can follow these steps: 1. Identify the center of the ellipse from the graph. The center is at \((-5, -1)\). 2. Determine the lengths of the semi-major and semi-minor axes. From the graph, the semi-major axis is vertical and has a length of 5, while the semi-minor axis is horizontal and has a length of 3. 3. Use the formula for the distance of the foci from the center for a vertical ellipse: \[ c = \sqrt{a^2 - b^2} \] where \(a\) is the semi-major axis length and \(b\) is the semi-minor axis length. \[ a = 5, \quad b = 3 \] \[ c = \sqrt{5^2 - 3^2} = \sqrt{25 - 9} = \sqrt{16} = 4 \] 4. The foci are located \(c\) units above and below the center along the major axis. The center is \((-5, -1)\), so the foci are at: \[ (-5, -1 + 4) = (-5, 3) \] \[ (-5, -1 - 4) = (-5, -5) \] None of the given options match this, so it appears there’s a mistake in the listed options.