Final answer:
To find the equation of the circle, we need to find the center and radius. The center is the midpoint between the given endpoints, and the radius is half the length of the diameter. The equation of the circle is (x - 1)² + (y - 2)² = 10.
Step-by-step explanation:
To find the equation of the circle, we need to find the center and radius of the circle. The center of the circle is the midpoint between the two given endpoints of the diameter, which can be found by taking the average of the x-coordinates and y-coordinates. The radius of the circle is half the length of the diameter. Once we have the center and radius, we can substitute them into the standard 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.
Step 1: Find the x-coordinate of the center:
(4 + (-2)) / 2 = 2 / 2 = 1
Step 2: Find the y-coordinate of the center:
(3 + 1) / 2 = 4 / 2 = 2
Step 3: Find the radius:
The length of the diameter is the distance between the two given endpoints, which can be found using the distance formula. In this case, the endpoints are (4, 3) and (-2, 1).
diameter = sqrt((4 - (-2))² + (3 - 1)²) = sqrt(6² + 2²) = sqrt(40) = 2√10
radius = (1/2) * 2√10 = √10
Step 4: Substitute the center and radius into the equation:
(x - 1)² + (y - 2)² = (√10)²
Therefore, the equation of the circle is (x - 1)² + (y - 2)² = 10.