Question

In Fig. if tangents PA and PB are drawn to a circle such that ∠APS = 30° and chord AC is drawn parallel to the tangent PB, then ∠ABC =

A. 60° 

B. 90° 

C. 30° 

D. None of these

Answer 1

Answer is C. 30°

Given: 

APB = 30° 

Property 1: If two tangents are drawn to a circle from one external point, then their tangent segments (lines joining the external point and the points of tangency on circle) are equal. 

Property 2: Sum of all angles of a triangle = 180° 

By property 1, 

PA = PB (tangent from P) 

And, 

∠PAB = ∠PBA [∵PA = PB] 

By property 2, 

∠PAB + ∠PBA + ∠APB = 180° 

⇒ ∠PAB + ∠PBA + 30° = 180° 

⇒ ∠PAB + ∠PBA = 180° - 30° 

⇒ ∠ PAB + ∠ PBA = 150° 

⇒ ∠ PBA + ∠ PBA = 150° [∵∠PAB = ∠PBA] 

⇒ 2∠PBA = 150°

⇒ ∠PBA = \(\frac{150^°}{2}\)

⇒ ∠PBA = 75° 

Now, 

∠PBA = ∠CAB = 75° [Alternate angles] 

∠PBA = ∠ACB = 75° [Alternate segment theorem] 

Again by property 2, 

∠CAB + ∠ACB + ∠CBA = 180° 

⇒ 75° + 75° + ∠CBA = 180°

⇒ 150° + ∠CBA = 180° 

⇒ ∠CBA = 180° - 150° 

⇒ ∠CBA = 30° 

Hence, ∠CBA = 30°