The 2 Tangent Parabolas

Calculus Level 3

On a vertical parabola \(P_1\), pick a point \(T\). Another vertical parabola \(P_2\) is tangent to \(P_1\) at \(T\). Spanning through all possible parabolas \(P_2\), what is the locus of the focus of \(P_2\)?

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