Question

From an external point P, two tangents PA and PB are drawn to the circle with centre O. Then the angle between OP and AB is_______.

Answer 1

PA and PB are tangents drawn from P to circle with centre O. Let OP and AB intersect at Q.

In ∆PAQ and ∆PBQ,

AP = BP (Length of tangents drawn from an external point to a circle are equal)

∠APQ = ∠BPQ (Tangents drawn from an external point to a circle are equally inclined to the segment joining the centre to that point)

PQ = PQ (Common)

∴ ∆PAQ $\cong$ ∆PBQ (SAS congruence axiom)

⇒ ∠AQP = ∠BQP (CPCT)

Also,

∠AQP + ∠BQP = 180º (Linear pair of angles)

⇒ 2∠AQP = 180º (∠AQP = ∠BQP)

⇒ ∠AQP = 90º

Therefore, OP is perpendicular to AB i.e. the angle between OP and AB is 90º.

From an external point P, two tangents PA and PB are drawn to the circle with centre O. Then the angle between OP and AB is __90º__.