Af = AC/2 and AE = AP/2 are obtained from the vertical longitude theorem of a circle.
EO 1//FO2
AO 1:AO2=AE:AF=3:2
AP:AC=3:2
From the tangent theorem, we know that Pb 2 = PC * PA
PB/PC=PA/PB
So Pb 2 = pa * (pa-AC) = pa * (pa-2pa/3) = pa 2/3.
PA/PB= root number 3
So PB/PC= root number 3
Choose B.