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!
Best Answer
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.