Help with Core 4 Parametric Equations

Question

I am currently working on parametric equations. I'm asked to express the parametric equations in Cartesian form and I can't since I can't make $t$ the subject for either equation.

$x=2t + t^2$

$y=2t^2 + t^3$

The question specifically says, by considering $\frac{y}{x}$, find a Cartesian form of the equations.

This lead me to:

$\frac{y}{x}=\frac{2t+t^2}{2+t}$

I'm struggling to see what I need to do from here. Any help is very much appreciated.

Answer 1

...so $\frac yx=t$ and you can substitute for $t$ to get the Cartesian form

Answer 2

HINT: from $$x=2t+t^2$$ we get $$t_{1,2}=-1\pm\sqrt{1+x}$$ and so you can eliminate $t$

Answer 3

Note that $$y=2t^2+t^3=t(2t+t^2)=tx$$ $$\implies t=\frac{y}{x}$$ So we have $$x=2t+t^2$$ $$x=2\cdot \frac{y}{x}+\left(\frac{y}{x}\right)^2$$ $$x^3-y^2-2xy=0$$

Hope you can complete it now.