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.