DETERMINE the equation of the circle that has the following three points on its circumference: A(1,5), B(3,7) and C(5,5).

Answer 1

(x-k)^2+ (y-h)^2 = r^2
so
(1-k)^2 + (5-h)^2 = r^2
(3-k)^2 + (7-h)^2 = r^2
(5-k)^2 + (5-h)^2 = r^2
Now before I go off and solve those for k,h and r, sketch a graph
Notice that two points, A and C are at the same height,5
The third point,B, is half way between them (3 is halfway between 1 and 5)
I conclude that the center of the circle is on x = 3
so
k = 3
Onward
(1-3)^2 + (5-h)^2 = r^2
(3-3)^2 + (7-h)^2 = r^2
(5-3)^2 + (5-h)^2 = r^2
that second equation is pretty easy now.
(7-h)^2 = r^2
the first and third are actually the same now that we know what k isso use the first and the second
4 + (5-h)^2 = r^2
(7-h)^2 = r^2

29 - 10 h + h^2 = r^2
49 -14 h + h^2 = r^2
----------------------
-20 +4 h = 0
h = 5
well I guess you can take it from there

Answer 2

Thank you!