Question

what are the foci of the ellipse $\frac{(x + 4)^{2}}{64}+\frac{(y - 7)^{2}}{94}=1$?
write your answer in simplified, rationalized form.
(, ) and (, )

Explanation:

Step1: Identify the form of the ellipse equation

The standard form of an ellipse is \(\frac{(x - h)^{2}}{b^{2}}+\frac{(y - k)^{2}}{a^{2}} = 1\) (\(a>b\)), where \((h,k)\) is the center. For the given equation \(\frac{(x + 4)^{2}}{64}+\frac{(y - 7)^{2}}{94}=1\), we have \(h=-4\), \(k = 7\), \(b^{2}=64\), \(a^{2}=94\).

Step2: Calculate \(c\)

Use the formula \(c=\sqrt{a^{2}-b^{2}}\). Substitute \(a^{2}=94\) and \(b^{2}=64\) into the formula: \(c=\sqrt{94 - 64}=\sqrt{30}\).

Step3: Determine the foci

Since the major axis is parallel to the \(y\)-axis (because \(a^{2}\) is under the \(y\) - term), the foci are \((h,k\pm c)\). Substitute \(h=-4\), \(k = 7\), and \(c=\sqrt{30}\) into the formula. The foci are \((-4,7+\sqrt{30})\) and \((-4,7 - \sqrt{30})\).

Answer:

\((-4,7+\sqrt{30})\) and \((-4,7 - \sqrt{30})\)