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A line in a given direction meeting two given lines

This type of question, asked 7 times: 14 Jun 2022, 25 Jan 2023, 10 Apr 2023, 29 Jan 2024, 9 Apr 2024, 21 Jan 2026 S2, 4 Apr 2026 S1. All 7 questions below, oldest first.

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  1. Question 1

    14 Jun 2022

    84
    Let a line having direction ratios 1, -4, 2 intersect the lines x−73=y−1−1=z+21\frac{x - 7}{3}=\frac{y - 1}{- 1}=\frac{z + 2}{1} and x2=y−73=z1\frac{x}{2}=\frac{y - 7}{3}=\frac{z}{1} at the point AA and BB. Then (AB)2(AB)^{2} is equal to

  2. Question 2

    25 Jan 2023

    70
    Consider the lines L1L_{1} and L2L_{2} given by
    L1:x−12=y−31=z−22L_{1}:\frac{x - 1}{2}=\frac{y - 3}{1}=\frac{z - 2}{2}
    L2:x−21=y−22=z−33L_{2}:\frac{x - 2}{1}=\frac{y - 2}{2}=\frac{z - 3}{3}. A line L3L_{3} having direction ratios 1,−1,−21, - 1, - 2, intersects L1L_{1} and L2L_{2} at the points PP and QQ respectively. Then the length of line segment PQPQ is

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  3. Question 3

    10 Apr 2023

    167
    Let the line x1=6−y2=z+85\frac{x}{1}=\frac{6 - y}{2}=\frac{z + 8}{5} intersect the lines x−54=y−73=z+21\frac{x - 5}{4}=\frac{y - 7}{3}=\frac{z + 2}{1} and x+36=3−y3=z−61\frac{x + 3}{6}=\frac{3 - y}{3}=\frac{z - 6}{1} at the points A and B respectively. Then the distance of the mid-point of the line segment ABAB from the plane 2x−2y+z=142x - 2y + z = 14 is

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  4. Question 4

    29 Jan 2024

    90
    A line with direction ratios 2,1,22,1,2 meets the lines x=y+2=zx = y + 2 = z and x+2=2y=2zx + 2 = 2y = 2z respectively at the point PP and QQ. if the length of the perpendicular from the point (1,2,12)(1,2,12) to the line PQPQ is ll, then l2l^{2} is

  5. Question 5

    9 Apr 2024

    61
    Let the line LL intersect the lines
    x−2=−y=z−1,2(x+1)=2(y−1)=z+1x - 2 = - y = z - 1,2(x + 1) = 2(y - 1) = z + 1
    and be parallel to the line x−23=y−11=z−22\frac{x- 2}{3}=\frac{y- 1}{1}=\frac{z- 2}{2}. Then which of the following points lies on LL ?

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  6. Question 6

    21 Jan 2026 S2

    58
    Let the line L1L_{1} be parallel to the vector −3i^+2j^+4k^- 3\widehat{i}+ 2\widehat{j}+ 4\widehat{k} and pass through the point (2,6,7)(2,6,7) and the line L2L_{2} be parallel to the vector 2i^+j^+3k^2\widehat{i}+\widehat{j}+ 3\widehat{k} and pass through the point (4,3,5)(4,3,5). If the line L3L_{3} is parallel to the vector −3i^+5j^+16k^- 3\widehat{i}+ 5\widehat{j}+ 16\widehat{k} and intersects the lines L1L_{1} and L2L_{2} at the points C and D , respectively, then ∣CD→∣2|\overrightarrow{CD}|^{2} is equal to :

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  7. Question 7

    4 Apr 2026 S1

    64
    A line with direction ratios 1,−1,21, - 1,2 intersects the lines x2=y3=z+13\frac{x}{2}=\frac{y}{3}=\frac{z + 1}{3} and x+1−1=y−21=z4\frac{x + 1}{- 1}=\frac{y - 2}{1}=\frac{z}{4} at the points P and Q , respectively. If the length of the line segment PQ is α\alpha, then 225α2225\alpha^{2} is equal to:

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