3D Geometry
Three Dimensional Geometry
nta_pyq_2025_jan
Grade 12
Question:
If the square of the shortest distance between the lines y-1 y+3 x-2 1 = 2 = z+3 -3 and x+1 2 = 4 = z+5 -5 is m n , where m, n are coprime numbers, then m + n is equal to :
Step-by-Step Solution
Key Concept: Apply the core result for lines and planes in three dimensions and simplify using the given constraints.
\to a = (2, 1, -3) (2) \to b = (-1, -3, -5) ∣^ ^ ^ ∣ i j k \to \to ∣ ∣ p \times q = ∣1 2 -3 ∣ ∣ ∣ ∣2 4 -5 ∣ ^ ^ = 2i - j \to \to ^ ^ ^ b - a = -3 i - 4 j - 2k \to \to \to \to |( b - a ) ⋅ ( p \times q )| Sd = - \to \to | p \times q | 2 = \sqrt5 2 4 ( Sd ) = 5 m = 4, n = 5 \Rightarrow m + n = 9
Correct Answer: 2