Question

if the angle between two tangent drawn from a external point pro a circle of radius a and center o, is 60 degree find the length of op

Answer 1

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 = OP/ OA

√ 3= OP/a

therefore, OP = √ 3a