Question

If the angle between two tangents drawn from an external point P to a circle of radius a and centre O, is 60°, then find the length of OP

Answer 1

Let PA and PB be the two tangents drawn to the circle with centre O and radius a such that ∠APB=60^°

In ∆OPB and ∆OPA

OB = OA = a (Radii of the circle)

∠OBP = ∠OAP=90^° (Tangents are perpendicular to radius at the point of contact)

BP = PA (Lengths of tangents drawn from an external point to the circle are equal)

So, ∆OPB ≌ ∆OPA (SAS Congruence Axiom)

∴ ∠OPB = ∠OPA=30^° (CPCT)

Now,

In ∆OPB

${\displaystyle {\sin 30}^{\circ }=\frac{\text{OB}}{\text{OP}}}$

${\displaystyle \Rightarrow \frac{1}{2}=\frac{a}{OP}}$

${\displaystyle \Rightarrow OP=2a}$

Thus the length of OP is 2a