- JEE Main 2026
- 08 April Evening
- Q15
- MCQ
- 4 marks
Shortest Distance Parallel Lines Q15
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Quick Summary
This is a two-stage question. First, the foot of perpendicular from (λ,2,3) on a given line is stated to be (1,μ,2); since that foot must lie ON the line, substituting pins down μ = 3 directly, and the perpendicularity condition then gives λ = 2. Second, with λ,μ known, the two lines in the actual question turn out to share the SAME direction ratios (2,3,6) — they are parallel, not skew — so the shortest distance between them is simply |AB × d|/|d| for any points A, B on the two lines. That computation gives exactly √146/7.
Question Replay
Let the foot of perpendicular from the point (λ, 2, 3) on the line (x−4)/1 = (y−9)/2 = (z−5)/1 be the point (1, μ, 2).
Then the distance between the lines (x−1)/2 = (y−2)/3 = (z+4)/6 and (x−λ)/2 = (y−μ)/3 = (z+5)/6 is equal to:
Options
Correct Answer
Correct Answer: Option (C) — √146/7
Why This Answer
The four options are deliberately close (12/7, √145/7, √146/7, √143/7) because they are all built from the same |AB × d| computation with small arithmetic variations — this rewards careful component-by-component cross-product work over pattern-matching. The setup step (finding λ and μ) is easy to rush past, but it is not optional: without the correct λ=2, μ=3, the second line’s point and direction would be wrong and every subsequent number would drift.
The most important structural observation is that after substituting λ, μ, the two lines in the main question share the identical direction (2,3,6) — this makes them parallel, and the correct tool is the point-to-line-style formula |AB×d|/|d|, not the general skew-lines shortest-distance formula (which uses a scalar triple product and would give zero here misleadingly, or requires an unnecessary extra check for parallelism first).
Core Concept
Foot of perpendicular lying on the line. If a stated foot point is claimed to lie on a given line, the fastest way to extract information is to substitute its coordinates into the line’s parametric form — this often fixes one unknown immediately (here μ), before even using the perpendicularity condition (which is then used only for the remaining unknown, λ).
Shortest distance between parallel lines. For two parallel lines with common direction d, passing through points A and B respectively, the shortest distance is d(A,B) = |AB × d| / |d| — the same formula as for a point-to-line distance, using the vector between any two points on the lines.
Step 2: (λ-1,-1,1)·(1,2,1)=0 ⇒ λ-1-2+1=0 ⇒ λ=2
Step 3: Line A: point (1,2,-4), dir (2,3,6). Line B: point (2,3,-5), dir (2,3,6) — PARALLEL
AB = (1,1,-1); AB × d = (9,-8,1); |AB×d|=√146; |d|=7
Distance = √146/7
Step-by-Step Solution
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Use the foot lying on the line to find μ. A general point on (x−4)/1=(y−9)/2=(z−5)/1 is (4+r, 9+2r, 5+r). Setting x=1 gives r=−3, so y=9−6=3 (and z=5−3=2, matching). So μ=3.
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Use perpendicularity to find λ. The vector from (λ,2,3) to the foot (1,3,2) is (1−λ, 1, −1). This must be perpendicular to the line’s direction (1,2,1):
(1-λ)(1) + (1)(2) + (-1)(1) = 0
1-λ+2-1 = 0 ⇒ λ = 2 -
Write the two lines with λ=2, μ=3. Line A: (x−1)/2=(y−2)/3=(z+4)/6, point A=(1,2,−4), direction (2,3,6). Line B: (x−2)/2=(y−3)/3=(z+5)/6, point B=(2,3,−5), same direction (2,3,6) — the lines are parallel.
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Compute AB and AB × d. AB = B−A = (1,1,−1):
AB × d = (1·6−(−1)·3, −(1·6−(−1)·2), 1·3−1·2) = (9, −8, 1)
|AB×d| = √(81+64+1) = √146 -
Divide by |d|. |d| = √(4+9+36) = √49 = 7. Distance = √146/7.
Final Answer
Final Answer: Option (C) — √146/7
The shortest distance between the two lines is exactly √146/7. In the exam, always check whether two lines are parallel BEFORE reaching for the general skew-lines distance formula — parallel lines use the simpler |AB×d|/|d| shortcut.
Key Facts to Remember
- If a stated point lies on a line, substitute its coordinates into the line’s parametric form to extract unknowns quickly.
- Foot-of-perpendicular condition: (P−N)·d=0, where d is the line’s direction.
- Two lines with proportional (here, identical) direction ratios are parallel.
- Shortest distance between parallel lines: |AB×d|/|d|, using any points A, B on each line.
Common Mistakes
Mistake 1: Jumping straight to a skew-lines shortest-distance formula (scalar triple product of AB, d1, d2) without first checking that d1 and d2 are actually identical here.
Fix: Always compare the two direction vectors first. If they are proportional, the lines are parallel and the much simpler |AB×d|/|d| applies directly.
Mistake 2: Skipping the foot-on-line substitution and trying to find both λ and μ from the perpendicularity condition alone, which is one equation in two unknowns.
Fix: Always use ALL given information — the fact that (1,μ,2) is stated to lie ON the line is itself an equation that fixes μ before perpendicularity is even needed.
FAQs
Q1. How do you find the shortest distance between two parallel lines in 3D?
Take one point on each line, form the vector AB between them, and compute |AB×d|/|d|, where d is the common direction vector. This is the perpendicular (shortest) distance between the two parallel lines.
Q2. What is the correct answer to this question?
The distance between the two lines is √146/7, option (3). This follows after first finding λ=2 and μ=3 from the foot-of-perpendicular condition given in the first part of the question.
Q3. How do you find λ and μ from the foot-of-perpendicular condition?
Since the given foot point (1, μ, 2) must lie on the line (x−4)/1=(y−9)/2=(z−5)/1, substitute to find the line parameter and read off μ directly. Then impose that the vector from (λ,2,3) to the foot is perpendicular to the line’s direction to solve for λ.
Q4. Why are the two lines in the second part parallel rather than skew?
Both lines are written with the same direction ratios (2,3,6) once λ and μ are substituted, so they are parallel by construction — the shortest-distance formula for parallel lines applies directly rather than the skew-lines formula.
Q5. How many marks is this question worth, and how long should it take?
It is a 4-mark MCQ with -1 negative marking. With both parts planned out, it takes about 3 minutes total.
Prerequisites
Before practising this question type, make sure the following are in place:
- Parametric form of a line and substituting a claimed point into it.
- Foot-of-perpendicular / dot-product-zero condition for perpendicularity.
- Recognising parallel lines from proportional direction ratios, and the |AB×d|/|d| shortest-distance formula.
Revise these from the Three Dimensional Geometry chapter, then attempt the related questions below.
Solved by Nishant Kumar Gupta for padholikhojee.in · JEE Main 2026 08 April Evening · Last updated: 2026-09-02 · Verified against the official question paper and answer key.