Locus of an Equation Bouncing off of another question I had here, I wondered what would be the equation of the locus of this equation: $$0 = b + ax - dy - cxy$$ Specifically $$0=xy-6y+6x-3$$ I would like a way to derive it. Also, if it is possible, may someone please give me a name for the function.

By itself, 
$$ (x-6)(y+6) = xy -6y+6x - 36 $$  You have
$$  0 = xy-6y+6x-3  $$
Subtract!
$$ (x-6)(y+6) = -33 $$
called a hyperbola, in the same way that $xy = 1$ is a hyperbola. Just moved a bit, stretched a bit. 
Let's check a point. My version has $-3 \cdot 11 = -33,$ so we can take $x=3, y=5.$ The original
$xy-6y+6x -3$ becomes $3 \cdot 5 - 6 \cdot 5 + 6 \cdot 3 - 3 = 15 -30+18 - 3=0$
