The general equation of a circle is given as #x^2+y^2+2gx+2fy+c=0# where #g, f, c# are constants .
Substituting the given three points one-by-one in the above equation,
1. #(6, -6)#
#36+36+12g-12f+c=0#
#implies 72+12g-12f+c=0#
2.#(3, -2)#
#9+4+6g-4f+c=0#
#implies 13+6g-4f+c=0#
3.#(7, -5)#
#49+25+14g-10f+c=0#
#implies 74+14g-10f+c=0#
subtracting 2. from 3.;
#74+14g-10f+c-(13+6g-4f+c)=0#
#implies 61+8g-6f=0# -------------------------( 4. )
subtracting 1. from 3.;
#74+14g-10f+c-(72+12g-12f+c)=0#
#implies 2+2g+2f=0#
#implies 1+g+f=0# ----------------------( 5. )
from 5., #g=-1-f# ---------------------( 6. )
substituting this value of #g# in 4.;
#61+8(-1-f)-6f=0#
#implies 61-8-8f-6f=0#
#implies 53=14f#
#implies color(red)(f=53/14)#
substituting this value of #f# in 6.;
#g=-1-53/14 = (-14-53)/14#
#implies color(red)(g=-67/14)#
substituting these values of #f# and #g# in any of the equations 1., 2., 3., to obtain the value of c.
Let's use 2.
#13-6*67/14-4*53/14+c=0#
#implies -216/7+c=0#
#implies color(red)(c=216/7)#
substituting these values of #g, f, c# in the general equation of a circle [#x^2+y^2+2gx+2fy+c=0#]
#x^2+y^2-67/7x+53/7y+216/7=0#
#implies color(red)(7x^2+7y^2-67x+53y+216=0)#
is the required equation of the circle.
