Vectors & 3D GeometrymediumPYQ · JEE Main · 5 Apr 2026 · Shift 1 (Morning)Free

Intersection of Two Lines, Distance² from Origin = 17 | JEE 2026

JEE Maths question with a full step-by-step solution.

Question
The square of the distance of the point of intersection of the lines r⃗=(i^+j^−k^)+λ(ai^−j^)\vec r=(\hat i+\hat j-\hat k)+\lambda(a\hat i-\hat j), a≠0a\ne0, and r⃗=(4i^−k^)+μ(2i^+ak^)\vec r=(4\hat i-\hat k)+\mu(2\hat i+a\hat k) from the origin is
A55
B1010
C1717correct
D2626
Solution
Step 1: Line 1: (1+aλ, 1−λ, −1)(1+a\lambda,\ 1-\lambda,\ -1); Line 2: (4+2μ, 0, −1+aμ)(4+2\mu,\ 0,\ -1+a\mu). Equate j^\hat j: 1−λ=0⇒λ=11-\lambda=0\Rightarrow\lambda=1. Step 2: Equate k^\hat k: −1=−1+aμ⇒aμ=0-1=-1+a\mu\Rightarrow a\mu=0; a≠0⇒μ=0a\ne0\Rightarrow\mu=0. Step 3: Equate i^\hat i: 1+a=4⇒a=31+a=4\Rightarrow a=3. Step 4: Line 1 with λ=1, a=3\lambda=1,\ a=3: (1+3, 1−1, −1)=(4,0,−1)(1+3,\ 1-1,\ -1)=(4,0,-1). Step 5: 42+02+(−1)2=16+0+1=174^2+0^2+(-1)^2=16+0+1=17. Correct answer: (3)
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