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Distance of a point from a plane measured along a given line

This type of question, asked 6 times: 11 May 2023 S2, 16 May 2023 S1, 14 May 2024 S1, 26 Apr 2025 S2, 15 Apr 2026 S2, 14 May 2026 S2. All 6 questions below, oldest first.

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  1. Question 1

    11 May 2023 S2

    123
    The distance of the point (1,−5,9)(1,-5,9) from the plane x−y+z=5x-y+z=5 measured along the line x=y=zx=y=z is

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  2. Question 2

    16 May 2023 S1

    120
    A line with positive direction cosines passes through the point P(2,−1,2)P(2,-1,2) and makes equal angles with coordinate axes. The line meets the plane 2x+y+z=92x+y+z=9 at point Q. Then the length of the line segment PQ equals

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  3. Question 3

    14 May 2024 S1

    130
    A line with positive direction cosines passes through the point P(2,−1,2)P(2,-1,2) and makes equal angles with the co-ordinate axes. The line meets the plane 2x+y+z=92x+y+z=9 at point QQ. The length of the line segment PQPQ equals

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  4. Question 4

    26 Apr 2025 S2

    142
    The distance of the point A(3,−4,5)A(3, - 4,5) from the plane 2x+5y−6z=162x+ 5y- 6z= 16 measured along the line x2=y1=z−2\frac{x}{2}=\frac{y}{1}=\frac{z}{- 2} is

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  5. Question 5

    15 Apr 2026 S2

    96
    The distance of the point (1,0,−3)(1, 0, -3) from the plane x−y−z=9x - y - z = 9 measured parallel to the line x−22=y+23=z−6−6\dfrac{x - 2}{2} = \dfrac{y + 2}{3} = \dfrac{z - 6}{-6} is

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  6. Question 6

    14 May 2026 S2

    93
    The distance of the point (4,5,3)(4, 5, 3) from the plane x+y+z=2x + y + z = 2 measured parallel to the line x=y=2zx = y = 2z is ..... units

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