Given a bilinear function $f(x,y) = a + bx + cy + dxy$, is there a closed form solution for a slope line passing through point $(x_0, y_0, f(x_0, y_0))$? It can exclude degenerate cases, e.g. $b = c = d = 0$.
Edit
I call $f$ bilinear due to my application domain, see this Wikipedia article, but the name is probably superfluous here, except that the domain can be restricted to $[0,1]\times[0,1]$. By slope line, I mean a curve that is everywhere orthogonal to the contours as defined in this Wikipedia article.
Divide through by $d;$ after that we have $d=1$ and new values for $b,c.$
Your curves are $$ xy + bx + c y = \; \; \mbox{something}, $$ OR $$ (x+c)(y+b) = \; \; \mbox{thing}, $$
The orthogonal family are also hyperbolas, $$ (x + c)^2 - (y + b)^2 = \; \; \mbox{something} \; \; \mbox{else} $$
At some point $(x,y),$ the gradient of the first function is $$ (y + b, x + c), $$ the gradient of the second function is $$ (2( x + c),-2(y+b) ). $$ So the dot product of the two gradients is zero.