Final answer:
The coordinates of the center of the ellipse, given by the equation x² + 36y² + 12x = 0, are found by completing the square for the x-terms and rewriting the equation in standard form, which results in (-6, 0).
Step-by-step explanation:
To find the coordinates of the center of the ellipse given by the equation x² + 36y² + 12x = 0, first, we need to complete the square for the x-terms.
The equation can be rewritten as:
x² + 12x + 36y² = 0
Add and subtract the square of half the coefficient of x, which is (12/2)² = 36, to complete the square:
x² + 12x + 36 - 36 + 36y² = 0
Rewrite this as:
(x + 6)² - 36 + 36y² = 0
To isolate the completed square, add 36 to both sides:
(x + 6)² + 36y² = 36
The equation now represents an ellipse in standard form, where the center is at the point (h, k) such that the equation looks like (x-h)²/a² + (y-k)²/b² = 1.
Comparing the standard form to our equation, we find the center of the ellipse is at (-6, 0).
So, the coordinates of the center of the ellipse are (-6, 0).