The general equation of a circle is given as x2+y2+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
⇒72+12g−12f+c=0
2.(3,−2)
9+4+6g−4f+c=0
⇒13+6g−4f+c=0
3.(7,−5)
49+25+14g−10f+c=0
⇒74+14g−10f+c=0
subtracting 2. from 3.;
74+14g−10f+c−(13+6g−4f+c)=0
⇒61+8g−6f=0 -------------------------( 4. )
subtracting 1. from 3.;
74+14g−10f+c−(72+12g−12f+c)=0
⇒2+2g+2f=0
⇒1+g+f=0 ----------------------( 5. )
from 5., g=−1−f ---------------------( 6. )
substituting this value of g in 4.;
61+8(−1−f)−6f=0
⇒61−8−8f−6f=0
⇒53=14f
⇒f=5314
substituting this value of f in 6.;
g=−1−5314=−14−5314
⇒g=−6714
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⋅6714−4⋅5314+c=0
⇒−2167+c=0
⇒c=2167
substituting these values of g,f,c in the general equation of a circle [x2+y2+2gx+2fy+c=0]
x2+y2−677x+537y+2167=0
⇒7x2+7y2−67x+53y+216=0
is the required equation of the circle.
