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\begin{aligned} x^{2}+(y-p)^{2} &=(y+p)^{2} \\ x^{2}+y^{2}-2 p y+p^{2} &=y^{2}+2 p y+p^{2} \\ x^{2}-2 p y &=2 p y \\ x^{2} &=4 p y \end{aligned}