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I want to find all points of intersection of the parametric curves

$$C_1:\begin{cases} x = t+1 \\ y = t^2 \\ \end{cases} $$ and $$C_2:\begin{cases} x = 3t+1 \\ y = t^2+1 \\ \end{cases} $$ I know how to find the two points of intersection by converting the parametric equations to Cartesian equations of the two parabolas $C_1:y=f_1(x)$ and $C_2:y=f_2(x)$ by solving for $x$ the equation $f_1(x)=f_2(x)$. My question is about the possibility of finding the intersection points of the two parametric curves without converting to cartesian equations and I thought that was possible by solving for $t$ the system : $$\begin{cases} t+1= 3t+1 \\ t^2= t^2+1 \\ \end{cases} $$ but obviously this system does not have a solution. Thank you for your help!

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    $\begingroup$ These are two slightly different problems. The former is about the points in $\mathbb{R}^2$ which belong to both curves. While the second one is about those points that belong to both curves and also are target of the same value of the same parameter $t$. $\endgroup$ Mar 29, 2016 at 20:08

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I believe you should represent each of the parametric variables as different entities and then solve, e.g.:$$\begin{cases} t_1+1= 3t_2+1 \\ t_1^2= t_2^2+1 \\ \end{cases}$$Solve for $t_1$ and $t_2$ here.


$t_1$ will give you the coordinates of $C_1$ and $t_2$ will give you the coordinates of $C_2$ at the points of intersection. Notes that both $C_1$ and $C_2$ should come out with the same $x$ and $y$ coordinates at the points of intersection.

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