Final answer:
The center of the circle given by the equation (x + 8)² + y² = 64 is (-8, 0). This is found by comparing the equation to the standard form of a circle's equation and identifying the coefficient of x and y within the parentheses.
Step-by-step explanation:
To find the center of a circle given by its equation, one must understand the standard equation of a circle centered at (h, k) with radius r: (x - h)² + (y - k)² = r². In the equation you've provided, (x + 8)² + y² = 64, the circle is a rearrangement of this standard form. By comparing the two, it's evident that the square of the radius of the circle is 64, which means that the radius is 8 units.
To identify the coordinates of the center, we look at the x and y components within the parentheses. The equation has (x + 8), which in the standard form correlates to (-h), indicating that h is -8. There is no subtraction or addition next to the y, suggesting k is 0. Thus, the center of the circle is (-8, 0).
Remembering the characteristics of a circle, all points on the circumference are equidistant from the center. This consistent distance is the radius, and for an ellipse, rather than a single center, there are two foci that satisfy a distance property. Despite these facts, in this question, we are dealing with a perfect circle, where the center point is the focus of our interest.