Question

if the angle between two tangents draw from an external point P to a circle of radius and a centre O, is 60degree,then find the lenght of OP

Answer 1

Answer:

Step-by-step explanation:

OA=a

OP=?

ANGLE BETWEEN TANGENTS=60°

Tangents are equally aligned to each other

=> <OPA=<OPB=30°

IN ∆OPA,

<POA=180°-90°-30°

=60°

Cos 60°=OA/OP

1/2 =a/OP

=> OP = 2a(Ans)

Answer 2

The length of OP is 2a units.

• Let PQ and PR be two tangents drawn from P to the circle with center 'O'.
• Given,

                ∠RPQ = 60°

• OP is angle bisector of ∠RPQ

                ⇒ ∠OPQ = ∠OPR = 30°

• Let 'a' be the radius of the circle.

                ⇒ OQ = OR = a

• Now, PQ⊥OQ and PR⊥OR    [∵ tangent is perpendicular to radius]
• From ΔOPQ,

                    $sin30=\frac{OQ}{OP}$

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

                ⇒ OP = 2a units