Question

if the angle between two tangent drawn from an external point P to a circle of radius a and center O is 60° then find the lenght 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 = AP/ OA

√ 3= AP/a

therefore, AP = √ 3a

hence, proved

I hope its help you
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