are you sure you've posted this correctly? if x=3t^{2} and y=3t^{2} then y = x, dy/dx = 1
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!!
This is a mess. First off. Is Q your curve given by x(t) and y(t)? Is it a point? I can't tell.
Next, the tangent to your curve is given by dy/dx and part of the whole idea of parametric formulation is that you can immediately say
dy/dx = (dy/dt) / (dx/dt)
Using your curve we get dy/dx = (6t^{2} / 6t) = t = sqrt(x/3), you found this, the hard way.
As for the rest of it. I really don't understand what it is you are trying to show. Tangent lines normal to the curve at Q? What does that mean?
I think the question is very clear on what is Q. I'm asked to prove that the tangent cuts the curve again at the point Q.
I don't know about the last part, which is why I'm asking. The question is as the book puts it. I myself don't understand how the tangent at a certain point in a curve can be a normal at another point in the same curve, which is why I'm posting the question and asking for help.