Control circle to hyperbola
Author: Daniela Velichová

Control circle to a hyperbola is the set of all points
symmetric to one focus of a hyperbola through its tangents.

Centre of the control circle determined by points 
symmetric to focus F2 through the hyperbola tangent lines
is the control circle with centre in the other focus F1 
and radius r = 2a = AB
equal to the doubled size of the hyperbola semi-major axis. 

Part of the control circle can be generated
by moving the tangent point T along the hyperbola c.

PROOF
TQ = TF2
r = QF1 = TF1 - TQ = TF1 - TF2 = 2a

F1 = (-1, -1)
F2 = (0.6351752609046653, 0.656466611651154)
c = Hyperbola(F1, F2, 0.8)
S = Midpoint(F1, F2)
o = Line(F1, F2)
A, B = Intersect(c, o)
T = Point(c)
t, b = Tangent(T, c)
k = OrthogonalLine(F2, t)
Q = Mirror(F2, t)
P = Intersect(t, k)
poly1, p, t_1, f2 = Polygon(T, F2, P)
poly2, q, t_2, p_1 = Polygon(T, P, Q)
r = Segment(F1, Q)
d = Circle(F1, Q)