If angle between two tangents drawn from an external point is 60 degree then find the length of OP

If angle between two tangents drawn from an external point is 60 degre

1.	If angle between two tangents drawn from an external point is 60 degree then find the length of OP
Answer» We know that tangent is always perpendicular to the radius at the point of contact.So, ∠OAP = 90We 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{tex}\\times{/tex}60° = 30°According to the angle sum property of triangle-In ∆AOP, ∠AOP + ∠OAP + ∠OPA = 180°{tex}\\Rightarrow{/tex}\xa0∠AOP + 90° + 30° = 180°{tex}\\Rightarrow{/tex}\xa0∠AOP = 60°So, in triangle AOPtan angle AOP = AP/ OA{tex}\\sqrt 3 = \\frac{{AP}}{a}{/tex}therefore, {tex}AP = \\sqrt 3 a{/tex}hence, proved