Josh starts at the point $A=(0,0)$ and is heading towards the point $P=(12,3)$ and he uses 3 hours to reach this point. Tommy is located at $B=(0,9)$. One hour after Josh starts to walk towards $P$, then Tommy starts to head towards point $P$ as well. I need to find a parametrization for the path to Tommy such that he and Josh ends up at $P$ at the same time.
Attempt: First we need to find where Josh is after one hour, I simply took $\mathbf{x}=(0,0)+t(12,3)$ where $t\in\mathbb{R}$. Since he uses three hours to reach $P$ then after one hour $(t=1/3)$ Josh is located at $\mathbf{x}=(0,0)+1/3(12,3)=(4,1)$. I'm not sure how to make sure that they end up at $P$ at the same time.
Here is one parametrization,
The path of Josh to point $P$ is given by,
$r_1(t) = (0,0) + \frac{t}{3}(12,3), 0 \leq t \leq 3$
Simplifying, $r_1 = (4t, t), 0 \leq t \leq 3$
As Tommy starts one hour later and reaches point $P$ in two hours, path of Tommy to point $P$ will be,
$r_2(t) = (0,9) + \frac{t-1}{2}(12 - 0,3 - 9), 1 \leq t \leq 3$
Simplifying, $ \ r_2(t) = (6t-6, 12-3t), 1 \leq t \leq 3$