The standard form for the equation of a circle is:
#(x - h)^2 + (y - k)^2 = r^2#
where #(x,y)# is any given point on the circle, #(h,k)# is the center, and r is the radius.
Use the standard form and the three given points to write 3 equations:
#(6 - h)^2 + (8 - k)^2 = r^2" [1]"#
#(5 - h)^2 + (4 - k)^2 = r^2" [2]"#
#(3 - h)^2 + (2 - k)^2 = r^2" [3]"#
Set the left side of equation [1] equal to the left side of equation [2]:
#(6 - h)^2 + (8 - k)^2 = (5 - h)^2 + (4 - k)^2" [4]"#
Set the left side of equation [1] equal to the left side of equation [3]:
#(6 - h)^2 + (8 - k)^2 = (3 - h)^2 + (2 - k)^2" [5]"#
Use the pattern, #(a - b)^2 = a^2 + 3ab + b^2# to expand the squares:
#36 - 12h + h^2 + 64 - 16k + k^2 = 25 - 10h + h^2 + 16 - 8k + k^2" [6]"#
#36 - 12h + h^2 + 64 - 16k + k^2 = 9 - 6h + h^2 + 4 - 4k + k^2" [7]"#
The #k^2 and h^2# terms cancel:
#36 - 12h + 64 - 16k = 25 - 10h + 16 - 8k" [8]"#
#36 - 12h + 64 - 16k = 9 - 6h + 4 - 4k" [9]"#
Collect the constant terms into a single term on the right:
#-12h - 16k = -10h - 8k - 59 " [10]"#
#-12h - 16k = -6h - 4k - 87" [11]"#
Collect all of the h terms into a single term on the right:
#-16k = 2h - 8k - 59 " [12]"#
#-16k = 6h - 4k - 87" [13]"#
Collect all of the k terms into a single term on the left:
#-8k = 2h - 59 " [14]"#
#-12k = 6h - 87" [15]"#
Divide equation [14] by -8 and equation [15] by -12
#k = -1/4h + 59/8 " [16]"#
#k = -1/2h + 87/12" [17]"#
Set the right side of equation [16] equal to the right side of equation [17]:
#-1/4h + 59/8 = -1/2h + 87/12" [18]"#
Solve for h:
#1/4h = 87/12 - 59/8#
#h = 87/3 - 59/2#
#h = -1/2#
Substitute #-1/2# for h into equation [17]:
#k = 1/4 + 87/12#
#k = 15/2#
Substitute the values of h and k into either equation, [1], [2], or [3]. I will use equation [1]:
#(6 - -1/2)^2 + (8 - 15/2)^2 = r^2#
#(13/2)^2 + (1/2)^2 = r^2#
#r^2 = 170/4 = 85/2#
The area of the circle is #pir^2#:
#A = (85pi)/2#
