JEE Main 2023 · 10 April · morning · Single correct · +4 / −1
The shortest distance between the lines (x + 2)/1 = y/(- 2) = (z - 5)/2 and (x - 4)/1 = (y - 1)/2 = (z + 3)/0 is :
- (a) 8
- (b) 7
- (c) 6
- (d) 9
Answer and solution
Answer (d) 9
Solution
Maths sir · 4 steps
Take a vector joining the two lines and project it onto the common perpendicular b⃗₁ × b⃗₂.
- Read off the lines. Line 1 passes through A(-2, 0, 5) with direction b⃗₁ = (1, -2, 2). Line 2 passes through B(4, 1, -3) with direction b⃗₂ = (1, 2, 0).
- Common perpendicular. b⃗₁ × b⃗₂ = ((-2)(0) - (2)(2), (2)(1) - (1)(0), (1)(2) - (-2)(1)) = (-4, 2, 4). Its magnitude is √(16 + 4 + 16) = 6.
d = (|AB⃗ · (b⃗₁ × b⃗₂)|)/(|b⃗₁ × b⃗₂|)
- Put in numbers. AB⃗ = (4 - (-2), 1 - 0, -3 - 5) = (6, 1, -8). Then AB⃗ · (-4, 2, 4) = -24 + 2 - 32 = -54.
- So. d = (|-54|)/6 = 9, which is option (d).
The idea to keep. For skew lines, the cross product of the directions is the common perpendicular. Project any joining vector onto it: dot product on top, length of the cross product below.
Answer from the NTA final answer key. Solution by PrepWiser.