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Point on the line through two given points at a given distance from one of them

The same question, asked 2 times: 31 Jan 2024*, 4 Apr 2024. Every wording below, oldest first.

Try each one first, then open its answer.

  1. Question 1

    31 Jan 2024

    178
    A line passes through A(4,−6,−2)A(4, - 6, - 2) and B(16,−2,4)B(16, - 2,4). The point P(a,b,c)P(a,b,c) where a,b,ca,b,c are non-negative integers, on the line ABAB lies at a distance of 21 units, from the point AA. The distance between the points P(a,b,c)P(a,b,c) and Q(4,−12,3)Q(4, - 12,3) is equal to

  2. Question 2

    4 Apr 2024

    76
    Let the point, on the line passing through the points P(1,−2,3)P(1, - 2,3) and Q(5,−4,7)Q(5, - 4,7), farther from the origin and at a distance of 9 units from the point PP, be (α,β,γ)(\alpha,\beta,\gamma). Then α2+β2+γ2\alpha^{2}+\beta^{2}+\gamma^{2} is equal to:

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