# Parametric Equation Question

• Dec 1st 2013, 06:49 PM
LimpSpider
Parametric Equation Question
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 (Speechless).

The full question is as follows:

Show that the tangent at the point P, with parameter t, on the curve $x=3t^2$, $y=3t^2$ has equation $y=tx-t^3$. Prove that this tangent will cut the curve again at the point $Q(\frac{3t^2}{4}, - \frac{t^3}{4})$. 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: $y=\frac{2x^{\frac{3}{2}}}{3\sqrt{3}}$ Implicitly differentiating, the gradient would be $\frac{\sqrt{x}}{\sqrt{3}}=t$ (From solving the first parametric equation for t). Solve then for tangent.

The gradient therefore is correct and shown as $y=tx-t^3$. Intersecting the gradient and the curve, $y=tx-t^3=\frac{2x^{\frac{3}{2}}}{3\sqrt{3}}$. Solving for $x$, I get $x=3t^2$ or $\frac{3t^2}{4}$ Since the former option is already taken by P, Q will have to take the other option. Using then $x=\frac{3t^2}{4}$, and substituting it into $y=\frac{2x^{\frac{3}{2}}}{3\sqrt{3}}$, I get the value of $y=\pm \frac{t^3}{4}$. 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 $t$, we can assume that the tangent at Q is $-\frac{1}{t}$. Here I'm stuck, since I'm not sure how to proceed. Can someone help me over this?

• Dec 1st 2013, 07:07 PM
romsek
Re: Parametric Equation Question
are you sure you've posted this correctly? if x=3t2 and y=3t2 then y = x, dy/dx = 1
• Dec 1st 2013, 08:39 PM
LimpSpider
Re: Parametric Equation Question
Ooops sorry, it should be $x=3t^2, y=2t^3$. All other information is correct. Sorry!
• Dec 1st 2013, 10:44 PM
romsek
Re: Parametric Equation Question
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 = (6t2 / 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?
• Dec 2nd 2013, 12:00 AM
LimpSpider
Re: Parametric Equation Question
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. :cool:
• Dec 2nd 2013, 12:15 AM
topsquark
Re: Parametric Equation Question
Attachment 29840

-Dan
• Dec 2nd 2013, 04:21 PM
LimpSpider
Re: Parametric Equation Question
Now I should be able to get my answer...I only plotted positive y values for the curve :P
• Dec 2nd 2013, 05:37 PM
johng
Re: Parametric Equation Question
Hi,
The first attachment solves your problem; fortunately, this was a text book problem so the main equation has a simple solution. The second attachment is a graph showing one of the two pairs of points P and Q which satisfy your problem.

Attachment 29846

Attachment 29847
• Dec 2nd 2013, 06:56 PM
johng
Re: Parametric Equation Question
Hi again,
Sorry, the derivation in my previous posting has a silly algebraic error. The conclusion is correct; I just mistyped. Here is a corrected version.

Attachment 29850