9x^2+9y^2-12x-6y-31=10

nice circle, what about it?
Are we completing the square to find its centre ?

9x^2+9y^2-12x-6y-31=10
9(x^2 - (4/3)x + ...) + 9(y^2 - (2/3)y + ... = 41
9(x^2 - (4/3)x + 4/9) + 9(y^2 - (2/3)y + 1/9 = 41 + 9(4/9) + 9(1/9)
9(x - 2/3)^2 + 9(y - 1/3)^2 = 46

take over