To find the reqd. area of the circumcircle of the #Delta ABC#, where #A(9,4), B(7,5), C(3,6)# , we have to first find out the radius of the circle, say #R.#
Suppose that, pt.#P(x,y)# is the circumcentre of #Delta ABC.#
Then, dist. #PA#=dist. #PB#=dist. #PC,# each #=R.#
#:. (PA)^2=(PB)^2=(PC)^2.# Using Dist. Formula, we get,
#(x-9)^2+(y-4)^2=(x-7)^2+(y-5)^2=(x-3)^2+(y-6)^2.#
#:. -18x+81-8y+16=-14x+49-10y+25=-6x+9-12y+36#
#:. -4x+2y+23=0..........(1),# [using first & second eqns.], &
#-8x+2y+29=0.............(2)#,[using second & third eqns.]
Then, #(1)-(2)# gives, #4x-6=0,# or, #x=3/2#, then by #(1), y=-17/2#
So, the circumcentre of #Delta ABC# is #P(3/2,-17/2).#
Hence, #R^2=(CP)^2=(3-3/2)^2+(6+17/2)^2=9/4+841/4=850/4=425/2,# giving the Area of Circumcircle of #DeltaABC#=#pi*R^2=425/2pi=212.5pi~=667.25#
