Let A={θ∈R:(13sinθ+23cosθ)2=13sin2(θ)+23cos2(θ)}. Then
A
A∩[0,π] is an empty set
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B
A∩[0,π] has excatly one point
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C
A∩[0,π] has excatly two point
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D
A∩[0,π] has more than two points
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Solution
The correct option is BA∩[0,π] has excatly one point (13sinθ+23cosθ)2=13sin2(θ)+23cos2(θ)⇒19sin2θ+49cos2θ+49sinθcosθ=13sin2θ+23cos2θ⇒49sinθcosθ=29sin2θ+29cos2θ⇒sin2θ=sin2θ+cos2θ⇒sin2θ=1⇒2θ=π2,5π2⇒θ=π4,5π4