Show that the locus of the foot of the perpendicular from the focus to the tangent of the parabola is ,the tangent to the vertex.
Given parabola is
Equation to any tangent to the parabola is
Equation to the line perpendicular to the above line through is
Solving the two equations,we get
which is the foot of the perpendicular from S on the tangent.
Hence the locus of P is which is the tangent at the vertex.