Try to find the ratio of two arcs

Question

O1 is the center of the big circle; O2 is the center of the small one.

Line(O2-P) is vertical to line(A-C).

Line(PQ):line(PB)=2:7

What's the ratio of arc(AC) and arc(CB)?

Answer 1

Let $PQ=2$ and $PB=7$: by Pythagoras' theorem we then get $BQ=\sqrt{53}$.

If $PH$ is an altitude of triangle $BPQ$, then: $$PH=PQ\cdot {PB\over BQ}={14\over\sqrt{53}}$$ and by similarity: $$QH={4\over\sqrt{53}}, \quad BH={49\over\sqrt{53}}, \quad HO_2=QO_2-QH={45\over 2\sqrt{53}}. $$ Triangle $APH$ is similar to $PHO_2$, hence: $$ AH={392\over45\sqrt{53}},\quad AO_2=AH+HO_2={53\sqrt{53}\over90} \quad\text{and}\quad PA={14\sqrt{53}\over45}. $$ Finally, $ABC$ is similar to $APO_2$ and $\displaystyle AB=AO_2+BO_2={49\sqrt{53}\over45}$, whence: $$ BC={49\over\sqrt{53}}, \quad AC={1372\over45\sqrt{53}}. $$ It follows that: $$ {AC\over BC}={28\over45}, \quad\text{and}\quad {\text{arc}(AC)\over \text{arc}(BC)}={\arctan(28/45)\over\arctan(45/28)}. $$