Final answer:
To find the center of the circle with diameter endpoints at (-6,4) and (6,1), we calculate the midpoint which is (0, 2.5). The center is also where the foci of a circle are located, as they coincide at a single point.
Step-by-step explanation:
The question asks us to use the endpoints of a diameter of a circle to determine additional information. The endpoints given are (-6,4) and (6,1). Since the diameter of a circle runs through the center and touches two points on the circumference, the midpoint of the diameter's endpoints will give us the center of the circle.
To find the midpoint, which is also the center of the circle, we average the x-coordinates and the y-coordinates separately. The midpoint, or center, is found as follows:
(x1 + x2) / 2 = (-6 + 6) / 2 = 0 / 2 = 0
(y1 + y2) / 2 = (4 + 1) / 2 = 5 / 2 = 2.5
Thus, the center of the circle is at (0, 2.5).
The diameter's length can be calculated using the distance formula. However, as the foci of a circle are mentioned in the provided information, it's important to note that for a circle, unlike an ellipse, the foci are coincident and located at the center of the circle. So, the foci would also be at (0, 2.5).