Let , #P(-3,2), Q(-8,-7) and R(-6,1)# be the points on the circle.
The general quadratic eqn. of circle is :
# S:x^2+y^2+2gx+2fy+c=0 ,.....to(1)#
where ,.#"Radius "r=sqrt(g^2+f^2-c) > 0" and center "C (-g,-f) #
The circle #S# contains points #P(-3,2),Q(-8,-7),R(-6,1) # respectively ,and satisfy #eqn.(1)#
#:.9+4-6g+4y+c=0=>-6g+4f+c=-13to(2)#
#64+49-16g-14f+c# =#0=>-16g-14f+c#=#-113to(3)#
#36+1-12g+2f+c=0=>-12g+2f+c=-37to(4)#
#(2)-(3)=>10g+18f=100=>5g+9f=50...........to(5)#
#(2)-(4)=>6g+2f=24=>3g+f=12=>f#=#12-3g ....(6)#
Subst. this value of #f# into #(5)#
#5g+9(12-3g)=50=>5g+108-27g=50#
#=>-22g=-58=>color(red)(g=29/11#
From #eqn(6)# ,we get
#f=12-3(29/11)=(132-87)/11=>color(red)(f=45/11#
Subst. values of #g and f# into # eqn(1)#
#-6(29/11)+4(45/11)+c=-13#
#=>-174+180+11c#=#-143=>11c=-143+174-180#
#=>11c=-149=>color(red)(c=-149/11#
From Eqn.#(1)# we get
# S:x^2+y^2+2(29/11)x+2(45/11)y-149/11=0#
Hence ,the eqn. of the circle is:
#11x^2+11y^2+58x+90y-149=0#
............................................................................................................
Note:
#(i)# center of the circle is : #C(-g,-f)=>C(-29/11,-45/11)#
#(ii)#Radius of the circle is : #r=sqrt((29/11)^2+(45/11)^2+149/11#
#=>r=sqrt(4505/121)#
