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

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answer is B.

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Detailed Solution

Given that the angle between two tangents drawn from an external point ‘P’ to a circle of radius ‘r’ and center O is 60⁰.
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We know that the length of the two tangents drawn from an external point to a circle are equal.
Therefore, PA = PB ………(1)
Given the angle between the two tangents is 60⁰.
∴ ∠ BPA = 60⁰ ………(2)
Let us apply SSS similarity criterion in ∆POB & ∆POA.

PB = PA [from eq (1)] OB = OA = r [radii of same circle] OP = OP [commonside] Δ POB ~ Δ POA [by SSS similarity criterion]

We know that, if two triangle are similar, then the corresponding of the two triangles are equal and the corresponding sides are in the same ratio.
So, ∠ BPO = ∠ APO ……….(3)

∠ BPA = ∠ BPO + ∠ APO 60 ° = ∠ BPO + ∠ BPO 60 ° = 2 ∠ BPO ∠ BPO = 60 ° 2 ∠ BPO = 30 ° = ∠ APO

We know that tangents are always perpendicular to the radius.
∴ ∠ PBO = ∠ PAO = 90⁰
Therefore, ∆POB & ∆POA are 30⁰- 60⁰- 90⁰.
We know that, 30⁰- 60⁰- 90⁰ triangles the hypotenuse is equal to twice the length of the shorter side, which is the side opposite to 30⁰.
In ∆POB, the side opposite to ∠BPO = 30⁰ is OB = r and the hypotenuse is OP.
OP = 2 OB
OP = 2 r
Hence, the length of OP is 2r units.
The correct option is (2).

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