The exhaustive range of values of θ in the interval (0,π) such that the points (3,5) and (3 sinθ, cosθ) lies on the same side of line x+y-1 = 0 is

L11:3+5−1=7>0L22>0⇒3sin⁡θ+cos⁡θ>1sin⁡(θ+π/6)>1/2π/6<θ+π/6<5π/6⇒0<θ<2π/3

The exhaustive range of values of θ in the interval (0,π) such that the points (3,5) and (3 sinθ, cosθ) lies on the same side of line x+y-1 = 0 is

L11:3+5−1=7>0L22>0⇒3sin⁡θ+cos⁡θ>1sin⁡(θ+π/6)>1/2π/6<θ+π/6<5π/6⇒0<θ<2π/3

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The exhaustive range of values of θ in the interval (0,π) such that the points (3,5) and ( $\sqrt{3}$ sinθ, cosθ) lies on the same side of line x+y-1 = 0 is

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$(0, \frac{2 π}{3})$

$(0, \frac{π}{4})$

$(0, \frac{π}{2})$

$(\frac{5 π}{6}, π)$

answer is A.

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$\begin{array}{l} L_{11} : 3 + 5 - 1 = 7 > 0 \\ L_{22} > 0 \Rightarrow \sqrt{3} \sin θ + \cos θ > 1 \\ \sin (θ + π / 6) > 1 / 2 \\ π / 6 < θ + π / 6 < 5 π / 6 \\ \Rightarrow 0 < θ < 2 π / 3 \end{array}$

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The exhaustive range of values of θ in the interval (0,π) such that the points (3,5) and (3 sinθ, cosθ) lies on the same side of line x+y-1 = 0 is

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The exhaustive range of values of θ in the interval (0,π) such that the points (3,5) and ( $\sqrt{3}$ sinθ, cosθ) lies on the same side of line x+y-1 = 0 is