Hello Everyone again! I'm trying to do this question and I'm rather stumped on the last part. Parametric Equation is turning into a nightmare for me .

The full question is as follows:

Show that the tangent at the point P, with parameter t, on the curve , has equation . Prove that this tangent will cut the curve again at the point . Find the coordinates of the possible positions of P if the tangent to the curve at P is the normal to the curve at Q.

To prove for the tangent, I solved (is there a better word) the parametric equation into a cartesian equation, which is: Implicitly differentiating, the gradient would be (From solving the first parametric equation for t). Solve then for tangent.

The gradient therefore is correct and shown as . Intersecting the gradient and the curve, . Solving for , I get or Since the former option is already taken by P, Q will have to take the other option. Using then , and substituting it into , I get the value of . Since the positive solution does not satisfy, the negative must be the correct answer.

Since the tangent to the curve at P is the normal to the curve at Q, and the tangent at P is , we can assume that the tangent at Q is . Here I'm stuck, since I'm not sure how to proceed. Can someone help me over this?

Thanks in advance!!