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Point where a line meets a plane, then a distance

This type of question, asked 4 times: 2021 Paper 1, 2021 Paper 17, 2021 Paper 22, 10 Apr 2023. All 4 questions below, oldest first.

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  1. Question 1

    2021 Paper 1

    69
    The distance of the point (1,1,9)(1,1,9) from the point of intersection of the line x−31=y−42=z−52\frac{x- 3}{1}=\frac{y- 4}{2}=\frac{z- 5}{2} and the plane x+y+z=17x+y+z= 17 is:

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  2. Question 2

    2021 Paper 17

    82
    The distance of the point P(3,4,4)P(3,4,4) from the point of intersection of the line joining the points Q(3,−4,−5)Q(3,-4,-5) and R(2,−3,1)R(2,-3,1), and the plane 2x+y+z=72x+y+z=7, is equal to

  3. Question 3

    2021 Paper 22

    83
    The square of the distance of the point of intersection of the line x−12=y−23=z+16\frac{x- 1}{2}=\frac{y- 2}{3}=\frac{z+ 1}{6} and the plane 2x−y+z=62x - y + z = 6 from the point (−1,−1,2)( - 1, - 1,2) is

  4. Question 4

    10 Apr 2023

    73
    Let PP be the point of intersection of the line x+33=y+21=1−z2\frac{x + 3}{3}=\frac{y + 2}{1}=\frac{1 - z}{2} and the plane x+y+z=2x + y + z = 2. If the distance of the point PP from the plane 3x−4y+12z=323x - 4y + 12z = 32 is qq, then qq and 2q2q are the roots of the equation

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