Answer:
Explanation:
The center of the circle is the midpoint of the diameter. To find the midpoint, we use the midpoint formula:
Midpoint = [(x1 + x2)/2, (y1 + y2)/2]
where (x1, y1) and (x2, y2) are the coordinates of the endpoints of the diameter.
Midpoint = [(3 + (-2))/2, (1 + 3)/2]
Midpoint = [1/2, 2]
So, the center of the circle is (1/2, 2).
The radius of the circle is half the length of the diameter. To find the length of the diameter, we use the distance formula:
Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)
where (x1, y1) and (x2, y2) are the coordinates of the endpoints of the diameter.
Distance = sqrt((-2 - 3)^2 + (3 - 1)^2)
Distance = sqrt(25 + 4)
Distance = sqrt(29)
So, the length of the diameter is sqrt(29).
The radius of the circle is half of sqrt(29), which is sqrt(29)/2.
Therefore, the equation of the circle is:
(x - 1/2)^2 + (y - 2)^2 = (sqrt(29)/2)^2
Simplifying this equation, we get:
(x - 1/2)^2 + (y - 2)^2 = 29/4
So, the equation of the circle with a diameter whose endpoints are (3, 1) and (-2, 3) is (x - 1/2)^2 + (y - 2)^2 = 29/4.