[Math] Number of normals to a parabola from a given point

conic sectionscoordinate systems

I know that from any point a maximum of three normals could be drawn to a parabola because the equation of normal is cubic.
But I want to know the condition on the point for the number of normals

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Claude's technique can be extended slightly to find the the points and normal lines given a particular point off of the parabola. Using this, I obtained an animation of some of the normals.

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Condition for Point on Parabola Axis to Draw 3 Distinct Normals – Conic Sections

[Math] The feet of three normals from a point to a parabola determine a triangle. Show that its centroid lies on the parabola’s axis.

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