JEE Main Mathematics: Shortest distance between two skew lines Practice Questions
9 free multiple-choice questions on Shortest distance between two skew lines (Analytical Geometry) for JEE Main Mathematics, each with the correct answer and a step-by-step explanation. Attempt a question, then reveal the solution to check your reasoning.
Practice thousands more JEE Main questions in the app →Q1. The shortest distance between the lines r = (î + 2ĵ + 3k̂) + λ(î − k̂) and r = (4î + ĵ + 5k̂) + μ(2ĵ − k̂) is
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Correct answer: A — 3
b₁ × b₂ = (1, 0, −1) × (0, 2, −1) = (2, 1, 2), |b₁ × b₂| = 3. a₂ − a₁ = (3, −1, 2); dot product = 6 − 1 + 4 = 9. SD = 9/3 = 3.
Q2. If the shortest distance between two non-parallel lines is zero, then the lines
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Correct answer: A — intersect
Shortest distance = 0 means the lines have a common point. Non-parallel lines with a common point intersect.
Q3. The shortest distance between the lines r = λ(î − ĵ + k̂) and r = (î + ĵ + kk̂) + μ(2î + ĵ − k̂) is √2. Then k is
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Correct answer: C — 1 or −3
b₁ × b₂ = (1, −1, 1) × (2, 1, −1) = (0, 3, 3), magnitude 3√2. (a₂ − a₁)·(b₁ × b₂) = (1, 1, k)·(0, 3, 3) = 3 + 3k. |3 + 3k|/(3√2) = √2 gives |1 + k| = 2. k = 1 or k = −3.
Q4. The distance between the parallel lines r = a₁ + λb and r = a₂ + μb is
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Correct answer: C — |b × (a₂ − a₁)|/|b|
The distance is |a₂ − a₁| sin θ, where θ is the angle between a₂ − a₁ and b. |a₂ − a₁| sin θ = |b × (a₂ − a₁)|/|b|.
Q5. The distance between the parallel lines x = y = z and x − 1 = y − 2 = z − 3 is
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Correct answer: D — √2
Common direction b = (1, 1, 1); a₂ − a₁ = (1, 2, 3). (a₂ − a₁) × b = (2 − 3, 3 − 1, 1 − 2) = (−1, 2, −1), magnitude √6. d = √6/√3 = √2.
Q6. The shortest distance between the x-axis and the line through (0, 3, 4) parallel to the y-axis is
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Correct answer: A — 4
b₁ × b₂ = î × ĵ = k̂. a₂ − a₁ = (0, 3, 4); dot with k̂ = 4. SD = 4/1 = 4.
Q7. The shortest distance between the lines (x − 1)/3 = (y + 1)/2 = (z − 2)/2 and (x − 2)/2 = (y − 1)/2 = (z − 5)/1 is
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Correct answer: B — 2
b₁ × b₂ = (3, 2, 2) × (2, 2, 1) = (−2, 1, 2), magnitude 3. a₂ − a₁ = (1, 2, 3); dot product = −2 + 2 + 6 = 6. SD = 6/3 = 2.
Q8. The shortest distance between the lines x/1 = (y − 1)/2 = (z + 1)/2 and (x − 2)/3 = (y + 1)/2 = (z − 3)/(−2) is
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Correct answer: C — 4
b₁ × b₂ = (1, 2, 2) × (3, 2, −2) = (−8, 8, −4), magnitude 12. a₂ − a₁ = (2, −2, 4); dot product = −16 − 16 − 16 = −48. SD = 48/12 = 4.
Q9. A cube has edge 2. The shortest distance between a body diagonal of the cube and an edge that does not meet it is
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Correct answer: C — √2
Diagonal: from (0, 0, 0) along (1, 1, 1); edge: from (2, 0, 0) along (0, 0, 1). b₁ × b₂ = (1, −1, 0), magnitude √2. (a₂ − a₁)·(b₁ × b₂) = (2, 0, 0)·(1, −1, 0) = 2. SD = 2/√2 = √2.
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