Locus of the foot of perpendicular from the focus to a tangent:

• Parabola y^2=4ax: locus is the tangent at the vertex (x=0), i.e., the y-axis.
• Ellipse x^2/a^2+y^2/b^2=1: locus is the auxiliary circle x^2+y^2=a^2.
• Hyperbola x^2/a^2-y^2/b^2=1: locus is the auxiliary circle x^2+y^2=a^2.

Locus of the midpoint of a focal chord:

• Parabola y^2=4ax: locus is y^2=2a(x-a), a parabola shifted right.

Locus of the intersection of perpendicular tangents:

• Circle: concentric circle (director circle).
• Ellipse: x^2+y^2=a^2+b^2 (director circle).
• Hyperbola: x^2+y^2=a^2-b^2 (director circle, exists only if a>b).
• Parabola: the directrix.