Final answer:
By solving the system of equations of the perpendicular bisectors and finding the intersection point, we can determine the center of the circle. We can find distance between the center and one of the given points to determine the radius. Using the center and radius, we can write the equation of the circle in the standard form (x-h)^2 + (y-k)^2 = r^2.
Step-by-step explanation:
To find the equation of a circle passing through the points A(2, -3), B(-1, 4), and C(5, 1), we need to find the center and radius of the circle. The center of the circle can be found by finding the intersection of the perpendicular bisectors of two of the chords AB and BC. Once we have the center, we can use the distance formula to find the radius. With the center and radius, we can write the equation of the circle in the form (x-h)^2 + (y-k)^2 = r^2, where (h, k) is the center and r is the radius.
First, find the slopes of the perpendicular bisectors of AB and BC: the perpendicular bisector of AB passes through the midpoint of AB which is ((2 + (-1))/2, (-3 + 4)/2) = (0.5, 0.5). The slope of AB is (4 - (-3))/(-1 - 2) = -7/3, so the slope of the perpendicular bisector of AB is 3/7. Similarly, the midpoint of BC is ((-1 + 5)/2, (4 + 1)/2) = (2, 2.5). The slope of BC is (1 - 4)/(5 - (-1)) = -1/2, so the slope of the perpendicular bisector of BC is 2.
Next, using the point-slope form of a line, we can write the equations of the perpendicular bisectors:
Perpendicular bisector of AB: y - 0.5 = 3/7(x - 0.5)
Perpendicular bisector of BC: y - 2.5 = 2(x - 2)
Solve the system of equations to find the center of the circle by setting the two equations equal to each other:
3/7(x - 0.5) = 2(x - 2)
3x/7 - 3/7 = 2x - 4
x(3/7 - 2) = -4 + 3/7
(5/7)x = -25/7
x = -5/7
Substitute the value of x into one of the equations to find y:
y - 0.5 = 3/7(-5/7 - 0.5)
y - 0.5 = -15/49 + 21/49
y = 30/49 + 21/49 + 0.5
y = 51/49 + 0.5
y = 51/49 + 24/49
y = 75/49
Therefore, the center of the circle is (-5/7, 75/49). To find the radius, use the distance formula between the center and one of the given points. Let's use point A: √[(2 - (-5/7))^2 + ((-3) - (75/49))^2] = √[(2 + 5/7)^2 + (-3 - 75/49)^2] = √[(14/7 + 5/7)^2 + (-3 - 75/49)^2] = √[(19/7)^2 + (-147/49)^2] = √[(361/49) + (10227/2401)] = √[(361*49 + 10227)/(49*49)] = √[17689/2401] = (√17689)/(√2401) = 133/49
Finally, with the center (-5/7, 75/49) and radius 133/49, we can write the equation of the circle as:
(x - (-5/7))^2 + (y - 75/49)^2 = (133/49)^2
Simplifying: (x + 5/7)^2 + (y - 75/49)^2 = 17689/2401
Therefore, the correct equation of the circle is x² + y² + (10/7)x - (150/49)y - 17689/2401 = 0 (option d).