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Plane through three given points, then a distance or foot of a perpendicular

This type of question, asked 3 times: 24 Jan 2023, 11 Apr 2023, 15 Apr 2023. All 3 questions below, oldest first.

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  1. Question 1

    24 Jan 2023

    80
    The distance of the point (7,−3,−4)(7, - 3, - 4) from the plane passing through the points (2,−3,1),(−1,1,−2)(2, - 3,1),( - 1,1, - 2) and (3,−4,2)(3, - 4,2) is :

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  2. Question 2

    11 Apr 2023

    160
    Let PP be the plane passing through the points (5,3(5,3, 0),(13,3,−2)0),(13,3, - 2) and (1,6,2)(1,6,2). For α∈N\alpha \in N, if the distances of the points A(3,4,α)A(3,4,\alpha) and B(2,α,a)B(2,\alpha,a) from the plane PP are 2 and 3 respectively, then the positive value of aa is

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  3. Question 3

    15 Apr 2023

    76
    Let the foot of perpendicular of the point P(3,−2,−9)P(3, - 2, - 9) on the plane passing through the points (−1,−2,−3),(9,3,4),(9,−2,1)( - 1, - 2, - 3),(9,3,4),(9, - 2,1) be Q(α,β,γ)Q(\alpha,\beta,\gamma). Then the distance of QQ from the origin is:

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