An ellipse has only one tangent line at any given point on its curve, but a point outside the ellipse can have two tangent lines connecting to it. The discussion clarifies that the original problem involves finding the equations of two tangent lines from the point (12, 3) to the ellipse defined by x^2 + 4y^2 = 36. Participants emphasize the importance of visualizing the problem by graphing the ellipse to understand the tangent lines better. The conversation also touches on the complexities of deriving tangent equations, suggesting the use of established formulas for clarity. Ultimately, the key takeaway is that while a point on the ellipse has one tangent, external points can connect to the ellipse with two tangents.