Forcing equation/Plane/Monomial parameter situation/Example

Let ${}X$ denote a plane (like $K^{2},\mathbb {R} ^{2},{\mathbb {A} }_{K}^{2}$) with coordinate functions ${}x$ and ${}y$. We consider an inhomogeneous linear equation of type

${}x^{a}t_{1}+y^{b}t_{2}=x^{c}y^{d}\,.$

The fiber of the solution set ${}T$ over a point ${}\neq (0,0)$ is one-dimensional, whereas the fiber over ${}(0,0)$ has dimension two (for ${}a,b,c,d\geq 1$). Many properties of ${}T$ depend on these four exponents.