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locus on a parabola

Source: Croatia 2000 4th Grade P1

May 9, 2021
conic sectionsgeometryconicsparabola

Problem Statement

Let P\mathcal P be the parabola y2=2pxy^2=2px, and let T0T_0 be a point on it. Point T0′T_0' is such that the midpoint of the segment T0T0′T_0T_0' lies on the axis of the parabola. For a variable point TT on P\mathcal P, the perpendicular from T0′T_0' to the line T0TT_0T intersects the line through TT parallel to the axis of P\mathcal P at a point T′T'. Find the locus of T′T'.
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