To find the equation of a circle passing through three given points, we can use the general equation of a circle, which is:
(x - h)^2 + (y - k)^2 = r^2
where (h, k) is the center of the circle, and r is the radius.
We can find the center and radius of the circle by solving a system of equations using the three given points. Here's how:
First, we can find the equations of the perpendicular bisectors of the line segments connecting the three points. The intersection of these bisectors will be the center of the circle.
The midpoint of the line segment connecting (1, 2) and (0, 3) is:
((1+0)/2, (2+3)/2) = (1/2, 5/2)
The slope of the line segment is:
(3-2)/(0-1) = -1
The slope of the perpendicular bisector is the negative reciprocal of the slope of the line segment, which is:
1/(-1) = -1
So the equation of the perpendicular bisector is:
y - (5/2) = (-1)(x - 1/2)
y = -x + 6
The midpoint of the line segment connecting (0, 3) and (-2, 7) is:
((0-2)/2, (3+7)/2) = (-1, 5)
The slope of the line segment is:
(7-3)/(-2-0) = -2
The slope of the perpendicular bisector is the negative reciprocal of the slope of the line segment, which is:
1/(-2) = -1/2
So the equation of the perpendicular bisector is:
y - 5 = (-1/2)(x + 1)
y = (-1/2)x + (11/2)
The intersection of these two lines is the center of the circle. Solving for x and y, we get:
-x + 6 = (-1/2)x + (11/2)
x = 1
Substituting x = 1 into either equation, we get:
y = -1 + 6 = 5
So the center of the circle is (1, 5).
To find the radius of the circle, we can use the distance formula between one of the given points and the center of the circle. Let's use (1, 2):
r^2 = (1 - 1)^2 + (2 - 5)^2
r^2 = 9
r = 3
Therefore, the equation of the circle passing through the points (1, 2), (0, 3), and (-2, 7) is:
(x - 1)^2 + (y - 5)^2 = 9