If the shortest distance between the line joining the points and , and the line is , then is equal to ______.
Show answer and solution
Answer: 18
Solution
The line through (1,2,3) and (2,3,4) has direction b₁ = (1,1,1).
The second line has direction b₂ = (2,-1,0).
b₁ × b₂ = |i j k; 1 1 1; 2 -1 0| = (0·1−1·(−1), 1·2−0·1, 1·(−1)−1·2) = (1, 2, −3).
|b₁ × b₂| = √(1+4+9) = √14.
Connect a point on line 1 to a point on line 2: take A = (1,2,3) on line 1 and B = (1,−1,2) on line 2.
AB = (1−1, −1−2, 2−3) = (0, −3, −1).
Shortest distance a = |AB · (b₁ × b₂)| / |b₁ × b₂|
= |(0)(1) + (−3)(2) + (−1)(−3)| / √14
= |0 − 6 + 3| / √14
= 3/√14.
So a² = 9/14, and 28a² = 28 × 9/14 = 18.
The second line has direction b₂ = (2,-1,0).
b₁ × b₂ = |i j k; 1 1 1; 2 -1 0| = (0·1−1·(−1), 1·2−0·1, 1·(−1)−1·2) = (1, 2, −3).
|b₁ × b₂| = √(1+4+9) = √14.
Connect a point on line 1 to a point on line 2: take A = (1,2,3) on line 1 and B = (1,−1,2) on line 2.
AB = (1−1, −1−2, 2−3) = (0, −3, −1).
Shortest distance a = |AB · (b₁ × b₂)| / |b₁ × b₂|
= |(0)(1) + (−3)(2) + (−1)(−3)| / √14
= |0 − 6 + 3| / √14
= 3/√14.
So a² = 9/14, and 28a² = 28 × 9/14 = 18.
[JEE Main 2023 · 25 Jan · Shift 2]