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Condition for two lines to intersect or be coplanar: find the unknown or the meeting point

This type of question, asked 8 times: 2021 Paper 12, 2021 Paper 17, 28 Jul 2022, 29 Jan 2023, 6 Apr 2023, 12 Apr 2023, 13 Apr 2023, 2 Apr 2026 S1. All 8 questions below, oldest first.

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  1. Question 1

    2021 Paper 12

    43
    The lines x=ay−1=z−2x = ay - 1 = z - 2 and x=3y−2=bz−2,(ab≠0)x= 3y- 2 =bz- 2,(ab\neq 0) are coplanar, if :

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  2. Question 2

    2021 Paper 17

    63
    For real numbers α\alpha and β\beta, if the straight lines x−α1=y−12=z−13\frac{x-\alpha}{1}=\frac{y-1}{2}=\frac{z-1}{3} and x−4β=y−63=z−73\frac{x-4}{\beta}=\frac{y-6}{3}=\frac{z-7}{3} intersect at a point which lies on the plane x+2y−z=8x+2y-z=8, then α−β\alpha-\beta is equal to:

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  3. Question 3

    28 Jul 2022

    165
    Let the lines x−1λ=y−21=z−32\frac{x - 1}{\lambda}=\frac{y - 2}{1}=\frac{z - 3}{2} and x+26−2=y+183=z+28λ\frac{x + 26}{- 2}=\frac{y + 18}{3}=\frac{z + 28}{\lambda} be coplanar and PP be the plane containing these two lines. Then which of the following points does NOT lies on P?

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  4. Question 4

    29 Jan 2023

    164
    If the lines x−11=y−22=z+31\frac{x - 1}{1}=\frac{y - 2}{2}=\frac{z + 3}{1} and x−a2=y+23=z−31\frac{x - a}{2}=\frac{y + 2}{3}=\frac{z - 3}{1} intersect at the point PP, then the distance of the point PP from the plane z=az = a is :

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  5. Question 5

    6 Apr 2023

    173
    If the lines x−12=2−y−3=z−3α\frac{x - 1}{2}=\frac{2 - y}{- 3}=\frac{z - 3}{\alpha} and x−45=y−12=zβ\frac{x - 4}{5}=\frac{y - 1}{2}=\frac{z}{\beta} intersect, then the magnitude of the minimum value of 8αβ8\alpha\beta is

  6. Question 6

    12 Apr 2023

    64
    Let the lines l1:x+53=y+41=z−α−2l_{1}:\frac{x + 5}{3}=\frac{y + 4}{1}=\frac{z - \alpha}{- 2} and l2:3x+l_{2}:3x + 2y+z−2=0=x−3y+2z−132y + z - 2 = 0 = x - 3y + 2z - 13 be coplanar. If the point P(a,b,c)P(a,b,c) on l1l_{1} is nearest to the point Q(−Q( - 4,−3,2)4, - 3,2), then ∣a∣+∣b∣+∣c∣|a| + |b| + |c| is equal to

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  7. Question 7

    13 Apr 2023

    169
    The line, that is coplanar to the line x+3−3=y−11=z−55\frac{x + 3}{- 3}=\frac{y - 1}{1}=\frac{z - 5}{5}, is

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  8. Question 8

    2 Apr 2026 S1

    63
    If the point of intersection of the lines x+13=y+a5=z+b+17\frac{x+ 1}{3}=\frac{y+a}{5}=\frac{z+b+ 1}{7} and x−21=y−b4=z−2a7\frac{x- 2}{1}=\frac{y-b}{4}=\frac{z- 2a}{7} lies on xyxy-plane, then the value of a+ba+b is :

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