The 2 Tangent Parabolas

Consider the vertical line passing through $T$ . The reflection of the vertical line about the line tangent to $P_1$ and $P_2$ through $T$ must pass through both the focus of $P_1$ and the focus of $P_2$ through the reflective property of parabolas that all light shone vertically in a parabola reflects through the focus.

But this means that focus of $P_1$ , $P_2$ , and the point $T$ are collinear. Since $P_1$ and $T$ are predefined, then $P_2$ 's locus must be the $\boxed{\text{line}}$ passing through the focus of $P_1$ and the point $T$ .