There are two overlapping circles. Point A is known (it lies on the y axis) and the y value is given. This point makes a tangent to a circle with an unknown centre point and radius (centre C on the diagram)Diagram. A tangent angle at an overlap is known at point B and diameter is specified for this circle.
From this how can you calculate the Centre point and radius of the unknown circle?
The two reference circles are a distraction. You have two points $A$ and $B$ on the unknown circle and the directions (after a little bit of computation) of the tangents to the unknown circle at those two points. Its center $C$ is at the intersection of the perpendiculars to those two tangent lines at the two points of tangency.
The slope of the tangent to either of the reference circles at the point $(x,y)$ is simply $-x/y$, and you can then find the slope of the tangent to the unknown circle via the formula for the tangent of a sum.