x^2 + y^2 - 12x + 10y + 61 = 36
x^2 - 12x + y^2 + 10y + 61 = 36
We want (x - 6)^2, so we need +36, so we do this:
x^2 - 12x + 36 - 36 + y^2 + 10y + 61 = 36
(x - 6)^2 - 36 + y^2 + 10y + 61 = 36
We want (y + 5)^2, so we need + 25, so we do this:
(x - 6)^2 - 36 + y^2 + 10y + 25 - 25 + 61 = 36
(x - 6)^2 - 36 + (y + 5)^2 - 25 + 61 = 36
clean up the numbers on the left
(x - 6)^2 + (y + 5)^2 - 36 - 25 + 61 = 36
(x - 6)^2 + (y + 5)^2 = 36
(x - 6)^2 + (y + 5)^2 = 6^2
It's a circle centered at (6,-5) with a radius of 6.