Many extraneous tangent lines
Exploring elliptic curves such as C = ImplicitCurve(x^3 - x - y^2). Then Tangent(C, P) for some point P on C, creates about 6 tangent lines, some of which appear to be duplicates. I desire the tangent line AT P, but the other lines given are tangents to C (passing through P), tangent at other points on C. The collection of lines is a single object, so I can't delete one or more lines without deleting all lines.
IF my curve were the graph of a function f, though it would be a pain since I am considering tangent lines at several points on C, I suspect I could define a new function, g, restricted to x-values within some small distance of the x-coordinate of my point P, and then construct Tangent(g, P). But I'm not sure if I can define a similar restriction of the implicit curve.