Regarding to scheme as follows, when i move Guiding Line (Purple Colored), Ball A (Ghost) will slide around Ball B on red Axis. For this reason, i need to find β angle to calculate the exact location of the Ball A (Ghost) moving around Ball B.
Ball A can never proceed behind of Ball B that's why max. positions should be tangent to Ball B. My constants are α (Guiding Line angle), da (distance of Ball A center to edge of Ball B) and r (Ball radius). (xa, ya) & (xb, yb) center coordinates of Ball A and Ball B are known. β should be described in terms of following constants.
If I understand correctly, you have an isosceles triangle (the angle "on top" is also $\beta$, where there's the description "Equals $\beta$ in max.Position").
That is, the length of the Guiding Line is also $da + r$.
Thus by the Law of Sines
$$ \frac{ 2r }{ \sin \alpha} = \frac{ da + r }{ \sin\beta } \implies \sin\beta = \frac{da + r}{ 2r } \sin\alpha \quad \text{, or}~~ \beta = \arcsin\bigl(\frac{da + r}{ 2r } \sin\alpha \bigr)$$ Seemingly this comes from a practical task so presumably all the numbers have values such that the arcsine (inverse of sine) can be taken without a hitch.