Final answer:
The equation of the ellipse with the center at the origin, one focus at (-12,0), and a semi-major axis length of 13 is x^2/13^2 + y^2/25 = 1
Step-by-step explanation:
For an ellipse with the center at the origin, the equation is given by:
x2/a2 + y2/b2 = 1
where a is the semi-major axis and b is the semi-minor axis. Since the ellipse has a center at the origin, we can assume that the foci lie along the x-axis and are equidistant from the origin.
Given that one focus is at (-12, 0) and the semi-major axis length is 13, we can determine that the value of a is 13.
Substituting the values into the equation, we have:
x2/132 + y2/b2 = 1
We can now find the value of b by using the distance formula between the center and one of the foci:
a2 = b2 + f2
where f is the distance between the center and one of the foci. In this case, f = 12.
Plugging in the values, we have:
132 = b2 + 122
169 = b2 + 144
b2 = 25
Therefore, the equation of the ellipse with the specified conditions is:
x2/132 + y2/25 = 1