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Line lying in a plane: find the unknowns

This type of question, asked 2 times: 2021 Paper 23, 29 Jun 2022. All 2 questions below, oldest first.

Try each one first, then open its answer.

  1. Question 1

    2021 Paper 23

    82
    Suppose the line x−2α=y−2−5=z+22\frac{x- 2}{\alpha}=\frac{y- 2}{- 5}=\frac{z+ 2}{2} lies on the plane x+3y−2z+β=0x+ 3y- 2z+\beta= 0. Then (α+β)(\alpha+\beta) is equal to

  2. Question 2

    29 Jun 2022

    162
    Let the line x−23=y+1−2=z+3−1\frac{x - 2}{3}=\frac{y + 1}{- 2}=\frac{z + 3}{- 1} lie on the plane px−qy+px - qy + z=5z = 5, for some p,q∈Rp,q \in R. The shortest distance of the plane from the origin is:

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