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#
