If the points where the lines $3x - 2y - 12 = 0$ and $x + ky + 3 = 0$ intersect both the coordinate axes are concyclic, then the number of possible real values of $k$ is:
Step-by-Step Solution
Key Concept: A geometric problem may have multiple valid solutions depending on how constraints are satisfied.
Multiple configurations are possible for three collinear points satisfying the given constraints. The two cases shown involve different placements: one with points at $(-1,0)$, $(4,0)$, and $(0,-6)$; another with points at $(-2,0)$, $(0,2)$, and $(4,0)$ and $(0,-6)$. Each configuration represents a valid geometric arrangement satisfying the collinearity and distance/coordinate conditions.
Correct Answer: 2