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Line given as the intersection of two planes: distance or foot of the perpendicular from a point

This type of question, asked 6 times: 2021 Paper 23, 2021 Paper 24, 2021 Paper 26, 26 Jul 2022, 24 Jan 2023, 30 Jan 2023. All 6 questions below, oldest first.

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  1. Question 1

    2021 Paper 23

    66
    The distance of the point (−1,2,−2)( - 1,2, - 2) from the line of intersection of the planes 2x+3y+2z=02x+ 3y+ 2z= 0 and x−2y+z=0x - 2y + z = 0 is:

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  2. Question 2

    2021 Paper 24

    72
    The distance of line 3y−2z−1=0=3x−z+43y- 2z- 1 = 0 = 3x-z+ 4 from the point (2,−1,6)(2, - 1,6) is:

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  3. Question 3

    2021 Paper 26

    61
    Let LL be the line of intersection of the two planes x+2y+z=6x+ 2y+z= 6 and y+2z=4y+ 2z= 4. If point P(α,β,γ)P(\alpha,\beta,\gamma) is the foot of perpendicular from (3,2,1)(3,2,1) on LL, then the value of 21(α+β+γ)21(\alpha+\beta+\gamma) equals:

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  4. Question 4

    26 Jul 2022

    74
    The length of the perpendicular from the point (1,−2,5)(1, - 2,5) on the line passing through (1,2,4)(1,2,4) and parallel to the line x+y−z=0=x−2y+3z−5x + y - z = 0 = x - 2y + 3z - 5 is:

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  5. Question 5

    24 Jan 2023

    168
    If the foot of the perpendicular drawn from (1,9,7)(1,9,7) to the line passing through the point (3,2,1)(3,2,1) and parallel to the planes x+2y+z=0x + 2y + z = 0 and 3y−z=33y - z = 3 is (α,β,γ)(\alpha,\beta,\gamma), then α+β+γ\alpha + \beta + \gamma is equal to

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  6. Question 6

    30 Jan 2023

    171
    Let a line LL pass through the point P(2,3,1)P(2,3,1) and be parallel to the line x+3y−2z−2=0=x−y+x + 3y - 2z - 2 = 0 = x - y + 2z2z. If the distance of LL from the point (5,3,8)(5,3,8) is α\alpha, then 3α23\alpha^{2} is equal to