15. If the angle between two tangents drawn from anexternal point P to

1.	15. If the angle between two tangents drawn from anexternal point P to a circle of radius a and centreO is 60o, then find the length of OP.
Answer» We know that tangent is always perpendicular to the radius at the point of contact. So, ∠OAP = 90 We know that if 2 tangents are drawn from an external point, then they are equally inclined to the line segment joining the centre to that point. So, ∠OPA = 12∠APB = 12×60° = 30° According to the angle sum property of triangle- In ∆AOP,∠AOP + ∠OAP + ∠OPA = 180°⇒∠AOP + 90° + 30° = 180°⇒∠AOP = 60° So, in triangle AOP tan angle AOP = AP/ OA √ 3= AP/a therefore, AP = √ 3a hence, proved

15. If the angle between two tangents drawn from anexternal point P to a circle of radius a and centreO is 60o, then find the length of OP.

Answer»

We know that tangent is always perpendicular to the radius at the point of contact.

So, ∠OAP = 90

We know that if 2 tangents are drawn from an external point, then they are equally inclined to the line segment joining the centre to that point.

So, ∠OPA = 12∠APB = 12×60° = 30°

According to the angle sum property of triangle-

In ∆AOP,∠AOP + ∠OAP + ∠OPA = 180°⇒∠AOP + 90° + 30° = 180°⇒∠AOP = 60°

So, in triangle AOP

tan angle AOP = AP/ OA

√ 3= AP/a

therefore, AP = √ 3a

hence, proved