Write the equation of the circle passing through points a(3; 1) , b(-2; 6) and c(-5; -3)

Stand form of a circle is (x-h)²+(y-k)²=r², (h,k) being the coordinates for the center of the circle, r is the radius.
use the given three points to find the three unknown constants h, k, and r
(3-h)²+(1-k)²=r²
(-2-h)²+(6-k)²=r²
(-5-h)²+(-3-k)²=r²

expand these three equations:
9-6h+h²+1-2k+k²=r²
4+4h+h²+36-12k+k²=r²
25+10h+h²+9+6k+k²=r²
subtract equation 1 from equation 2, you get: 10h-10k=-30, h-k=-3
subtract equation 2 from equation 3, you get: 6h+18k=6, h+3k=1

subtract one form the other: 4k=4, k=1
h=1-3k=-2
find r²: (3+2)²+(1-1)²=r², (r=5)
so the equation is (x+2)²+(y-1)²=5²

6(2x + 5y)6 (2x) = 12x6 (5y) = 30y12x + 30y