#### 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

#### Solution

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

`sin 30^@ = "OB"/"OP"`

`=> 1/2 = a/(OP)`

`=> OP= 2a`

Thus the length of OP is 2a

Is there an error in this question or solution?

Solution 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 Concept: Tangent to a Circle.