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Question

Let A(α,β), B(α2,β), C(α,β2) are three ditinct points which are at same distance from origin. Then the sum of all possible value of ′θ′ such that (sinθ,cosθ) is equidistant to any of these points taken pairwise is where (0≤θ≤π2)

A
π4
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B
3π4
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C
π2
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D
0
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Solution

The correct option is B 3π4 Since points A,B,C are at the same distance from origin ⇒ α2+β2=α4+β2=α2+β4, Possible values for α,β : 0,1,−1 with ′0′ including distinct points are not possible ∴ A(−1,−1), B(1,−1) C(−1,1) is the only possible case here Let P(sinθ,cosθ) PA=PB⇒ (sinθ+1)2+(cosθ+1)2=(sinθ−1)2+(cosθ+1)2 ⇒ sinθ=0⇒θ=0∘ BP=PC ⇒ (sinθ−1)2+(cosθ+1)2=(sinθ+1)2+(cosθ−1)2 ⇒ sinθ=cosθ⇒θ=π4 AP=PC ⇒ (sinθ+1)2+(cosθ+1)2=(sinθ+1)2+(cosθ−1)2 ⇒ cosθ=0 ⇒ θ=π2 Hence required sum is =3π4

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