Question

Two circles touch internally at point P and a chord AB of the circle of longer radius intersects the other circle in C and D. Which of the following holds good?

(a) ∠CPA = ∠DPB 

(b) 2 ∠CPA = ∠CPD 

(c) ∠APX = ∠ADP 

(d) ∠BPY = ∠CPD + ∠CPA

Answer 1

Answer : (a) ∠CPA = ∠DPB 

In the bigger circle, ∠APX = ∠ABP 

In the smaller circle, ∠CPX = ∠PDC {Angles in alternate segment are equal.}

⇒ ∠APX + ∠CPA = ∠CPX = ∠PDC 

⇒ ∠ABP + ∠CPA = ∠PDC (∵ ∠APX = ∠ABP) 

⇒ ∠ABP + ∠CPA = ∠DBP + ∠DPB (ext. ∠ theorem in ∆ PDB)

∠ABP + ∠DPB 

⇒ ∠CPA = ∠DPB.