Parametric, Polar and Vector-Valued Functions · Challenge problem

Two particles move in the plane for $t\ge0$: particle $A$ has position $\langle t,\;t^2\rangle$ and particle $B$ has position $\langle 2t-1,\;3t-2\rangle$. Their paths cross at two points, and at one of them the particles actually collide (are there at the same time). Find the distance between the two crossing points.