Ellipse

Parabola

These constructions of the loci 
of the points that are equidistant 
from P and a conic centered 
at the origin are very similar to the 
construction of the loci that are 
equidistant from a point and a circle
(using the normal line
to the conic at C).

However, the equation is 
much more complicated to find.

Hyperbola

d: x = -1
I: (1, 0)
I': (-1, 0)
op1 = false
op2 = false
c = If(op1, Ellipse[I, I', 2], If[op2, Parabola[I, d], Hyperbola[I, I', 0.5]])
C = Point(c)
t = Tangent(C, c)
P = Point(xAxis)
r = LineBisector(P, C)
s = OrthogonalLine(C, t)
D = Intersect(r, s)
lugar = Locus(D, C)
anima = 0 (slider from 0 to 1)
O: (0, 0)
op3 = false
f = Segment(D, C)
de = Distance(ClosestPoint[c, O], O)
g = Segment(D, P)