To determine the equation of the ellipse, let's follow the necessary steps:
1. Identify the Values:
- The center of the ellipse is at the origin \((0,0)\).
- The vertex along the major axis is at \((13,0)\). This means the semi-major axis length \(a\) is 13.
- The focus is at \((12,0)\), giving us the distance from the center to the focus, \(c\), which is 12.
2. Relationship Between \(a\), \(b\), and \(c\):
- In an ellipse, the relationship between the semi-major axis \(a\), the semi-minor axis \(b\), and the distance to the focus \(c\) is given by the equation:
[tex]\[
c^2 = a^2 - b^2
\][/tex]
3. Calculate \(a^2\) and \(c^2\):
- Given \(a = 13\):
[tex]\[
a^2 = 13^2 = 169
\][/tex]
- Given \(c = 12\):
[tex]\[
c^2 = 12^2 = 144
\][/tex]
4. Solve for \(b^2\):
- Using the relationship \(c^2 = a^2 - b^2\):
[tex]\[
144 = 169 - b^2
\][/tex]
- Rearrange to solve for \(b^2\):
[tex]\[
b^2 = 169 - 144 = 25
\][/tex]
5. Equation of the Ellipse:
- The standard form of the equation of an ellipse centered at the origin is:
[tex]\[
\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1
\][/tex]
- Substitute the values \(a^2 = 169\) and \(b^2 = 25\):
[tex]\[
\frac{x^2}{13^2} + \frac{y^2}{5^2} = 1
\][/tex]
This matches the third option provided in the question. Therefore, the equation of the ellipse is:
[tex]\[
\boxed{\frac{x^2}{13^2} + \frac{y^2}{5^2} = 1}
\][/tex]
