To write the equation of the ellipse, we need the center, the lengths of the semi-major and semi-minor axes, and the orientation of the ellipse. From the problem statement, we have:
- Center: \((h, k) = (5, 3)\)
- Vertex: \((13, 3)\)
- Focus: \((5 + 2\sqrt{14}, 3)\)
Step 1: Determine the orientation.
Since the vertex and focus have the same \(y\)-coordinate (3), the ellipse is horizontal.
Step 2: Calculate the lengths.
The distance from the center to the vertex gives us \(a\):
- Distance from the center to the vertex: \[ a = 13 - 5 = 8 \]
The distance from the center to the focus gives us \(c\):
- Distance from the center to the focus: \[ c = (5 + 2\sqrt{14}) - 5 = 2\sqrt{14} \]
Step 3: Use the relationship between \(a\), \(b\), and \(c\).
For ellipses, the relationship is: \[ c^2 = a^2 - b^2. \] Substituting the known values: \[ (2\sqrt{14})^2 = 8^2 - b^2. \]
Calculating \(c^2\) and \(a^2\): \[ 4 \cdot 14 = 56, \] \[ 64 = 8^2. \]
Now substitute: \[ 56 = 64 - b^2, \] which leads to: \[ b^2 = 64 - 56 = 8. \]
Step 4: Write the equation of the ellipse.
In general form for a horizontally oriented ellipse centered at \((h, k)\): \[ \frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1. \]
Substituting in the values of \(h\), \(k\), \(a^2 = 64\), and \(b^2 = 8\): \[ \frac{(x - 5)^2}{64} + \frac{(y - 3)^2}{8} = 1. \]
Final Answer: \[ \frac{(x - 5)^2}{64} + \frac{(y - 3)^2}{8} = 1. \]
Therefore, the correct option is: Start Fraction left-parenthesis x minus 5 right-parenthesis squared over 64 End Fraction plus Start Fraction left-parenthesis y minus 3 right-parenthesis squared over 8 End Fraction equals 1.