To determine the center and the radius of a circle from its equation, we need to rewrite the equation in standard form, which is (x - h)^2 + (y - k)^2 = r^2.
The given equation is x^2 + 2x + y^2 - 2y + 14 = 0.
To rewrite it in standard form, we complete the square for both variables x and y.
For x, we add (2/2)^2 = 1 to both sides of the equation:
x^2 + 2x + 1 + y^2 - 2y + 14 = 1
(x + 1)^2 + y^2 - 2y + 15 = 1
For y, we add (-2/2)^2 = 1 to both sides of the equation:
(x + 1)^2 + (y - 1)^2 + 15 = 1 + 1
(x + 1)^2 + (y - 1)^2 + 15 = 2
Now, the equation is in standard form: (x + 1)^2 + (y - 1)^2 = 2 - 15
(x + 1)^2 + (y - 1)^2 = -13
Comparing it to (x - h)^2 + (y - k)^2 = r^2, we can determine that the center of the circle is at (-1, 1), and the radius is sqrt(-13).
Since the radius is the square root of a negative number, it means that the given equation does not represent a circle in the Euclidean coordinate system.