The standard form for the equation of a circle is:
r^2 = (x - h)^2 + (y - k)^2
where, (x,y) is any point on the circle, (h, k) is the center point, and r is the radius.
Using the two points we can write two equations:
r^2 = (4 - h)^2 + (5 - k)^2
r^2 = (3 - h)^2 + (7 - k)^2
Because r^2 = r^2, the right sides of these equations are equal:
(4 - h)^2 + (5 - k)^2= (3 - h)^2 + (7 - k)^2
I will expand the squares using the pattern (a - b)^2 = a^2 - 2ab + b^2:
16 - 8h + h^2 + 25 - 10k + k^2= 9 - 6h + h^2 + 49 - 14k + k^2
Combine like terms:
4k= 2h + 17
We substitute the point (h, k) into the given equation:
k = 7/9h + 7
Solve these two equation for h then k:
4(7/9h + 7)= 2h + 17
28/9h + 28 = 2h + 17
10/9h = -11
h = -99/10
4k= 2(-99/10) + 17
4k= -198/10 + 170/10
4k= -28/10
k = -7/10
Check:
-7/10 = 7/9(-99/10) + 7
-7/10 = -77/10 + 70/10
This checks
Use one of the equations of the circle and the center (-99/10, -7/10) to find r:
r^2 = (4 - -99/10)^2 + (5 - -7/10)^2
r^2 = (40/10 - -99/10)^2 + (50/10 - -7/10)^2
r^2 = 19321/100 + 3249/100
r^2 = 22570/100
r = sqrt(22570)/10
The equation of the circle is:
(sqrt(22570)/10)^2 = (x - -99/10)^2 + (y - -7/10)^2
