The general equation of a circle of radius $u$ that is tangent to the $x$-axis is $(x-h)^2+(y-u)^2=u^2$. Our strategy, then, is to find the radical line of this variable circle with the original circle, and then find the condition such that the radical line of both circles is tangent to them as well.
Start with the radical line:
rad = ((x - h)^2 + (y - u)^2 - u^2) - (x^2 + (y - 1)^2 - 1) // Simplify
h^2 - 2 h x - 2 (-1 + u) y
Then, determine the condition so that this line is tangent to the unit circle. In algebraic terms, we want the resulting quadratic polynomial after elimination to be a perfect square:
Solve[Discriminant[x^2 + (y - 1)^2 - 1 /. First[Solve[rad == 0, y]], x] == 0, h]
{{h -> 0}, {h -> 0}, {h -> -2 Sqrt[u]}, {h -> 2 Sqrt[u]}}
where we get two trivial solutions and a solution for both the right and left parts of the plane.
Now, we can visualize:
Graphics[{Circle[{0, 1}, 1],
MapIndexed[{ColorData[97, #2[[1]]], #1} &,
Table[Circle[{2 Sqrt[u], u}, u], {u, 1/20, 2, 1/20}]]},
Axes -> True]
