# parametric equations help

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• May 13th 2010, 08:53 AM
Tweety
parametric equations help
The curve with parametric equations $x = 3t^{2} , y = 4t^{3}$ along with the tangent at the point with parameter t = 1

find the equation if this tangent and the coordinates of the point where it meets the curve again.

$\frac{dx}{dt} = 6t$

$\frac{dy}{dt} = 12t^{2}$

$\frac{dy}{dx} = \frac{12t^{2}}{6t} = 2t$

when t = 1

$x = 3 , y = 4 , \frac{dy}{dx} = 2$

equation of tangent = $y -4 = 2(x-3)$

$y = 2x-2$

however to find the coordinates of where the curve crosses the tangent again, would I not just be able to sub in the values of x and y into the tnagent equation?

$4t^{3} = 2(3t^{2})-2$

$4t^{3} = 6t^{2} - 2$ can't seem to solve from here,

any help appreicated.

thanks
• May 13th 2010, 10:52 AM
Grandad
Hello Tweety
Quote:

Originally Posted by Tweety
The curve with parametric equations $x = 3t^{2} , y = 4t^{3}$ along with the tangent at the point with parameter t = 1

find the equation if this tangent and the coordinates of the point where it meets the curve again.

$\frac{dx}{dt} = 6t$

$\frac{dy}{dt} = 12t^{2}$

$\frac{dy}{dx} = \frac{12t^{2}}{6t} = 2t$

when t = 1

$x = 3 , y = 4 , \frac{dy}{dx} = 2$

equation of tangent = $y -4 = 2(x-3)$

$y = 2x-2$

however to find the coordinates of where the curve crosses the tangent again, would I not just be able to sub in the values of x and y into the tnagent equation?

$4t^{3} = 2(3t^{2})-2$

$4t^{3} = 6t^{2} - 2$ can't seem to solve from here,

any help appreicated.

thanks

Your working is perfect up to here. Good job!

What you need to realise now is that the tangent at $t=1$ effectively cuts the curve in two coincident points where $t = 1$.

So $t=1$ will be a repeated root of your final equation, which is more simply written as:
$2t^3-3t^2+1=0$
In other words $(t-1)^2$, or $(t^2-2t+1)$, is a factor of the LHS of:
$2t^3 - 3t^2 +1 =0$
So we get:
$(t^2-2t+1)(2t+1)=0$
(Check it out: multiply out the brackets and you get $2t^3-3t^2+1$.)

So the other value of $t$ is ...?

I'm sure you can take it from here.

Grandad