Final answer:
To find the coordinates of point O, the center of the circle, we use the midpoint formula on the diameter endpoints R(3a, 2b - 1) and S(a - 6, 4b + 5), resulting in point O having coordinates (2a - 3, 3b + 2).
Step-by-step explanation:
To find the coordinates of point O, which is the center of circle O with diameter RS, we need to determine the midpoint of the line segment RS with endpoints R(3a, 2b - 1) and S(a - 6, 4b + 5). The midpoint formula is ((x1 + x2)/2, (y1 + y2)/2), where (x1, y1) is the first endpoint and (x2, y2) is the second endpoint.
Substituting the coordinates of R and S into the midpoint formula gives us the coordinates of O as ((3a + (a - 6))/2, (2b - 1 + (4b + 5))/2), which simplifies to ((4a - 6)/2, (6b + 4)/2). After simplifying, you get the coordinates of point O as (2a - 3, 3b + 2).