[Math] Locus of a point where two normals meet

conic sections

Another exam question,

"Find the locus of a the point such that two of the normals drawn through it to the parabola $y^2=4ax$ are perpendicular to each other."

Does the locus mean the point of intersection of the two normals? I attempted to try to this by using the implicit derivative of the parabola and the locus as (x1,y1). Since its given as they meet but I can't get points of intersection.

Can someone help me out please?

Best Answer

Animation of the specified locus, for $a = 1/2$:

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