with their Inversions

The hyperbolas are computed with this
c(t) = [a * cosh(t), b * sinh(t)]
Note: (x/a)^2 - (y/b)^2 = 1 ("implicit equation")
The point & tangent construction is shown:
Intersect the ray from the left focal point with the
circle. Connect this point S with the right focal
point F. The symmetry line between S and F is the
hyperbola tangent where the ray meets the hyperbola.

Inversion(vector) = (radius/|vector|)^2 * vector
The inverted curves are lemniscates