Ok, what ideas have you had so far?
(2) You are trying to find a crunode: Crunode - Wikipedia, the free encyclopedia.
Note that crunodes occur at points where dy/dx has the indeterminant form 0/0. So get dy/dx, find the value of t for which dy/dx is indeterminant and then substitute this value of t into x = x(t) and y = y(t). (You know in advance that the point is a crunode but you should test it anyway).
Well, what else can it be? If the derivative were not indeterminant, then it would have one "determinant" value and that would be the slope of the tangent line.
But mr fantastic was answering a different question, "why is 0/0 not "infinite?"(giving a vertical tangent). The answer to that is that limits such as , all involve fractions of the form "0/0" but different limits and so different "slopes". To have an infinite limit your fraction must be of the form "a/0" where a is not 0.