Draw the circles as in the left diagram. We can safely assume x = 1, since we need to only find the ratio y/x.

First we find the distance between the points where X and Y touch the line. This will also serve as the perpendicular distance between the center of X and the vertical radius of Y.
(y-1)^2 + h^2 = (y+1)^2
-2y + h^2 = 2y
h^2 = 4y
h = 2sq(y).

Onto the second diagram!
Notice that the height of the center of the highest circle above the line is y+1+1 = y+2. We draw a perpendicular line leading left from the center of the left Y toward the vertical line of symmetry, forming another right triangle. It has width 2sq(y), height y+2-y = 2, and hypotenuse 2y.
2^2 + (2sq(y))^2 = (2y)^2
4 + 4y = 4y^2
1 + y = y^2
y^2 - y - 1 = 0
y = y/x = (1 + ¬/5)/2. This is the familiar phi, the golden ratio.