Please answer my question (4)?

Dear student,

G i v e n l e n g t h o f c o m m o n c h o r d A B = 12 c m L e t t h e r a d i u s o f t h e c i r c l e w i t h c e n t r e O i s O A = 10 c m R a d i u s o f c i r c l e w i t h c e n t r e P i s A P = 8 c m F r o m t h e f i g u r e, O P ⊥ A B \Rightarrow A C = C B ∴ A C = 6 c m (S i n c e A B = 12 c m) I n Δ A C P, A P^{2} = P C^{2} + A C^{2} [B y P y t h a g o r a s t h e o r e m] \Rightarrow 8^{2} = P C^{2} + 6^{2} \Rightarrow P C^{2} = 64 - 36 = 28 P C = 2 \sqrt{7} c m C o n s i d e r Δ A C O, A O^{2} = O C^{2} + A C^{2} [B y P y t h a g o r a s t h e o r e m] \Rightarrow 10^{2} = O C^{2} + 6^{2} \Rightarrow O C^{2} = 100 - 36 = 64 \Rightarrow O C = 8 c m F r o m t h e f i g u r e, O P = O C + P C = 8 + 2 \sqrt{7} c m H e n c e, t h e d i s \tan c e b e t w e e n t h e c e n t r e s i s (8 + 2 \sqrt{7}) c m . R e g a r d s