1. **State the problem:** Find the domain $D$ of the function $f(x,y) = \frac{x}{y}$ and represent it geometrically. 2. **Recall the domain rule for fractions:** The denominator cannot be zero because division by zero is undefined. 3. **Apply the rule:** For $f(x,y) = \frac{x}{y}$, the denominator is $y$. So, $y \neq 0$. 4. **Write the domain:** The domain $D$ is all pairs $(x,y)$ such that $y \neq 0$. 5. **Geometric representation:** In the $xy$-plane, this is the entire plane except the line $y=0$ (the $x$-axis). **Final answer:** $$D = \{(x,y) \in \mathbb{R}^2 : y \neq 0\}$$