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Angle between the line (x+1)/1 = (y-1)/2 = (z-2)/2 and the plane 2x - y + (root k)z + 4 = 0 has sine 1/3: find k

The same question, asked 2 times: 19 Apr 2025 S1, 14 May 2026 S1. Every wording below, oldest first.

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  1. Question 1

    19 Apr 2025 S1

    131
    If the angle θ\theta between the line x+11=y−12=z−22\frac{x+ 1}{1}=\frac{y- 1}{2}=\frac{z- 2}{2} and the plane 2x−y+λz+4=02x-y+\sqrt{\lambda}z+ 4 = 0 is such that sin⁡θ=13\sin\theta=\frac{1}{3}, then λ+1=\lambda+ 1 =

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  2. Question 2

    14 May 2026 S1

    93
    If the angle θ\theta between the line x+11=y−12=z−22\dfrac{x + 1}{1} = \dfrac{y - 1}{2} = \dfrac{z - 2}{2} and the plane 2x−y+kz+4=02x - y + \sqrt{k}z + 4 = 0 is such that cos⁡θ=83\cos\theta = \dfrac{\sqrt{8}}{3}. Then k=k =

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