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Straight Lines and Pair of Straight Lines question

2020 · 2 Sep · Shift 2 · Q34
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Straight Lines and Pair of Straight Lines question

2020 · 2 Sep · Shift 2 · Q34

JEE MainMathematicsStraight Lines and Pair of Straight LinesMCQ+4 / −1
The set of all possible values of θ\thetaθ in the interval (0, π\piπ) for which the points (1, 2) and (sin θ\thetaθ, cos θ\thetaθ) lie on the same side of the line x + y = 1 is :
  1. A
    (0,π4)\left( {0,{\pi \over 4}} \right)(0,4π​)
  2. B
    (0,3π4)\left( {0,{{3\pi } \over 4}} \right)(0,43π​)
  3. C
    (π4,3π4)\left( {{\pi \over 4},{{3\pi } \over 4}} \right)(4π​,43π​)
  4. D
    (0,π2)\left( {0,{\pi \over 2}} \right)(0,2π​)
View written solutionFree

Correct answer: D

  1. Equation of the line and side-test

    The given line is x+y=1.x+y=1.x+y=1. Define f(x,y)=x+y−1.f(x,y)=x+y-1.f(x,y)=x+y−1. Then:

    • if f(x,y)>0f(x,y)>0f(x,y)>0, the point lies on one side of the line,
    • if f(x,y)<0f(x,y)<0f(x,y)<0, it lies on the other side,
    • if f(x,y)=0f(x,y)=0f(x,y)=0, it lies on the line.

  2. Check the point (1,2)(1,2)(1,2)

    f(1,2)=1+2−1=2>0.f(1,2)=1+2-1=2>0.f(1,2)=1+2−1=2>0.

    So (1,2)(1,2)(1,2) lies on the side where x+y−1>0x+y-1>0x+y−1>0.

  3. Condition for (sin⁡θ,cos⁡θ)(\sin\theta,\cos\theta)(sinθ,cosθ) to be on the same side

    We need f(sin⁡θ,cos⁡θ)>0.f(\sin\theta,\cos\theta)>0.f(sinθ,cosθ)>0. Hence, sin⁡θ+cos⁡θ−1>0\sin\theta+\cos\theta-1>0sinθ+cosθ−1>0 sin⁡θ+cos⁡θ>1.\sin\theta+\cos\theta>1.sinθ+cosθ>1.

  4. Simplify the trigonometric inequality

    Use sin⁡θ+cos⁡θ=2sin⁡(θ+π4).\sin\theta+\cos\theta=\sqrt{2}\sin\left(\theta+\frac{\pi}{4}\right).sinθ+cosθ=2​sin(θ+4π​). So the condition becomes 2sin⁡(θ+π4)>1\sqrt{2}\sin\left(\theta+\frac{\pi}{4}\right)>12​sin(θ+4π​)>1 sin⁡(θ+π4)>12.\sin\left(\theta+\frac{\pi}{4}\right)>\frac{1}{\sqrt{2}}.sin(θ+4π​)>2​1​.

    Now, sin⁡u>12\sin u>\frac{1}{\sqrt{2}}sinu>2​1​ when u∈(π4,3π4).u\in\left(\frac{\pi}{4},\frac{3\pi}{4}\right).u∈(4π​,43π​).

    Let u=θ+π4.u=\theta+\frac{\pi}{4}.u=θ+4π​. Then θ+π4∈(π4,3π4).\theta+\frac{\pi}{4}\in\left(\frac{\pi}{4},\frac{3\pi}{4}\right).θ+4π​∈(4π​,43π​). Subtracting π4\frac{\pi}{4}4π​, θ∈(0,π2).\theta\in\left(0,\frac{\pi}{2}\right).θ∈(0,2π​).

  5. Match with the options

    This corresponds to (0,π2),\left(0,\frac{\pi}{2}\right),(0,2π​), which is Option D.

  6. Comparison with stored answer

    Stored correct answer: D

    Our derived answer: D

    So they agree.

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