Final answer:
To find the equation of a circle in standard form given two points A (6,6) and C (3,2), first determine the center by finding the midpoint of AC and then calculate the radius using the distance formula. The resultant equation is (x - 4.5)^2 + (y - 4)^2 = 6.25.
Step-by-step explanation:
To find the equation of the circle in standard form given two points, A and C, we need to identify the center and radius of the circle. The center of the circle will be the midpoint of segment AC and the radius will be the distance from the center to either point A or C.
First, calculate the midpoint (h, k) of AC:
h = (x1 + x2) / 2, k = (y1 + y2) / 2
h = (6 + 3) / 2, k = (6 + 2) / 2
h = 4.5, k = 4
Then we find the radius (r) using the distance formula:
r = √((x2 - x1)² + (y2 - y1)²).
Substituting the midpoint coordinates for h and k, and the coordinates of A or C, we get:
r = √((6 - 4.5)² + (6 - 4)²).
r = √(2.25 + 4).
r = √6.25.
r = 2.5.
Hence, the equation of the circle in standard form with center (4.5, 4) and radius 2.5 is:
(x - 4.5)² + (y - 4)² = 2.5².
(x - 4.5)² + (y - 4)² = 6.25.