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Perpendicular distance of a point from a plane

This type of question, asked 7 times: 4 May 2023 S2, 11 May 2023 S1, 10 May 2024 S2, 25 Apr 2025 S1, 11 Apr 2026 S1, 16 Apr 2026 S1, 14 May 2026 S1. All 7 questions below, oldest first.

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  1. Question 1

    4 May 2023 S2

    102
    If the points (1,−1,λ)(1,-1,\lambda) and (−3,0,1)(-3,0,1) are equidistant from the plane 3x−4y−12z+13=03x-4y-12z+13=0, then the sum of all possible values of λ\lambda is

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  2. Question 2

    11 May 2023 S1

    116
    The perpendicular distance of the origin from the plane 2x+y−2z−18=02x + y - 2z - 18 = 0 is

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  3. Question 3

    10 May 2024 S2

    126
    The perpendicular distance of the origin from the plane x−3y+4z−6=0x-3y+4z-6=0 is

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  4. Question 4

    25 Apr 2025 S1

    118
    The length of the foot of the perpendicular from the point (1,32,2)\left( 1,\frac{3}{2},2 \right) to the plane 2x−2y+4z+17=02x- 2y+ 4z+ 17 = 0 is

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  5. Question 5

    11 Apr 2026 S1

    93
    If the points (1,1,μ)(1, 1, \mu) and (−3,0,1)(-3, 0, 1) are equidistant from the plane r⃗⋅(3i^+4j^−12k^)+13=0\vec{r}\cdot(3\hat{i} + 4\hat{j} - 12\hat{k}) + 13 = 0, then the value of μ\mu are

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  6. Question 6

    16 Apr 2026 S1

    93
    If the product of the distances of the point (1, 2, 3) from the origin and the plane 2x−3y+z+k=02x - 3y + z + k = 0 is 7, then the value of k is

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  7. Question 7

    14 May 2026 S1

    95
    The distance of the point 2i^+j^−k^2\hat{i} + \hat{j} - \hat{k} from the plane rˉ⋅(i^−2j^+4k^)=9\bar{r}\cdot(\hat{i} - 2\hat{j} + 4\hat{k}) = 9, is

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