If the shortest between the lines
and
is 6, then the square of sum of all possible values of is
and
is 6, then the square of sum of all possible values of is
Show answer and solution
Answer: 384
Solution
## Solution
The direction vectors are b₁ = (2,3,4) and b₂ = (3,4,5).
Cross product b₁ × b₂:
b₁ × b₂ = |i j k; 2 3 4; 3 4 5| = (15-16, 12-10, 8-9) = (-1, 2, -1)
|b₁ × b₂| = √(1+4+1) = √6
Vector connecting points on the two lines:
Point on L₁: A = (-√6, √6, √6)
Point on L₂: B = (λ, 2√6, -2√6)
AB = (λ+√6, 2√6-√6, -2√6-√6) = (λ+√6, √6, -3√6)
Shortest distance formula:
d = |AB · (b₁ × b₂)| / |b₁ × b₂|
AB · (-1, 2, -1):
= -1·(λ+√6) + 2·(√6) + (-1)·(-3√6)
= -(λ+√6) + 2√6 + 3√6
= -λ - √6 + 5√6
= -λ + 4√6
Setting distance = 6:
|-λ + 4√6| / √6 = 6
|-λ + 4√6| = 6√6
Case 1: -λ + 4
The direction vectors are b₁ = (2,3,4) and b₂ = (3,4,5).
Cross product b₁ × b₂:
b₁ × b₂ = |i j k; 2 3 4; 3 4 5| = (15-16, 12-10, 8-9) = (-1, 2, -1)
|b₁ × b₂| = √(1+4+1) = √6
Vector connecting points on the two lines:
Point on L₁: A = (-√6, √6, √6)
Point on L₂: B = (λ, 2√6, -2√6)
AB = (λ+√6, 2√6-√6, -2√6-√6) = (λ+√6, √6, -3√6)
Shortest distance formula:
d = |AB · (b₁ × b₂)| / |b₁ × b₂|
AB · (-1, 2, -1):
= -1·(λ+√6) + 2·(√6) + (-1)·(-3√6)
= -(λ+√6) + 2√6 + 3√6
= -λ - √6 + 5√6
= -λ + 4√6
Setting distance = 6:
|-λ + 4√6| / √6 = 6
|-λ + 4√6| = 6√6
Case 1: -λ + 4
[JEE Main 2023 · 24 Jan · Shift 2]