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 given by the graph, let’s determine the necessary characteristics. The equation of an ellipse centered at \((h, k)\) with a vertical major axis is: \[ \frac{(x-h)^2}{b^2} + \frac{(y-k)^2}{a^2} = 1 \] where \(a > b\), and the foci are located at \((h, k \pm c)\), where \(c = \sqrt{a^2 - b^2}\). From the graph, the center of the ellipse appears to be at \((-5, 0)\). The major axis is vertical, and it spans from below \(y = 6\) to above \(y = -6\), giving a total length of 12, so \(a = 6\). The minor axis spans from \(x = -7\) to \(x = -3\), giving a total length of 4, so \(b = 2\). Calculate \(c\): \[ c = \sqrt{6^2 - 2^2} = \sqrt{36 - 4} = \sqrt{32} = 4\sqrt{2} \] The foci are at: \[ (h, k \pm c) = (-5, 0 \pm 4\sqrt{2}) \] Approximating \(4\sqrt{2} \approx 5.656\), the foci are approximately: \[ (-5, 5.656) \text{ and } (-5, -5.656) \] However, given options require estimating simpler values, so the closest answer is \((-5, 4)\) and \((-5, -4)\). Thus the correct choice is: \((-5, 4)\) and \((-5, -4)\)