Final answer:
To find the equation of the circle through (1, 2), (3, 4), and (5, 6), we calculate the center as the midpoint of the diameter formed by the points and the radius as the distance from the center to any of the points. The resulting equation is (x - 3)² + (y - 4)² = 8, which does not match the provided options; however, option (a) is the closest.
Step-by-step explanation:
To find the equation of the circle passing through the points (1, 2), (3, 4), and (5, 6), we will use the general equation of a circle, which is (x - h)² + (y - k)² = r², where (h, k) is the center of the circle, and r is the radius.
Firstly, we notice that the points lie on a line with a slope of 1, since the changes in y are the same as the changes in x for the sequential points. This meant that any circle touching all three points would be a circle with this line as its diameter. For such a circle, the midpoint of the diameter is the center of the circle. The midpoint of (1, 2) and (5, 6) is ((1+5)/2, (2+6)/2) which simplifies to (3, 4).
The radius can be found by calculating the distance from the center to any of the three points. Using the distance formula √((x2 - x1)² + (y2 - y1)², we find that the radius from the center (3, 4) to the point (1, 2) is √((1-3)² + (2-4)²) which is √(4 + 4) = √8 = 2√2.
Therefore, the equation of the circle is (x - 3)² + (y - 4)² = (2√2)². Simplifying the right side of the equation we get 4 × 2 = 8, so the equation of the circle in standard form is (x - 3)² + (y - 4)² = 8. However, this equation does not match any of the options provided, which suggests there may have been a mistake. If the options are based on an error, option (a) with the value 4 seems to be the closest approximation to the correct answer.