3D Geometry
Three Dimensional Geometry
nta_pyq_2025_jan
Grade 12
Question:
Let a = ^i + 2 ^j + k^ and b = 2 ^i + 7 ^j + 3 k^ . Let L 1 : r = (- ^i + 2 ^j + k^ ) + \lambda a , \lambda \in R and L 2 : r = ( j^ + k^ ) + \mu b , \mu \in R be two lines. If the line L 3 passes through the point of intersection of L 1 and L 2 , and is parallel to a + b , then L 3 passes through the point :
(5, 17, 4)
(2, 8, 5)
(8, 26, 12)
(-1, -1, 1)
Step-by-Step Solution
Key Concept: Apply the core result for lines and planes in three dimensions and simplify using the given constraints.
L : \to ^ ^ ^ ^ ^ ^ r = (- i + 2 j + k) + \lambda( i + 2 j + k) 1 (3) \Rightarrow \to ^ ^ ^ r = (\lambda - 1) i + 2(\lambda + 1) j + (\lambda + 1)k \to ^ ^ ^ ^ ^ L2 : r = ( j + k) + \mu(2 i + 7 j + 3k) \to ^ ^ ^ \Rightarrow r = 2\mu i + (1 + 7\mu) j + (1 + 3\mu)k For point of intersection equating respective components \Rightarrow \lambda - 1 = 2\mu 2(\lambda + 1) = 1 + 7\mu \lambda + 1 = 1 + 3\mu We get \Rightarrow \lambda = 3 and \mu = 1 \to \to ^ ^ ^ \Rightarrow a + b = 3 i + 9 j + 4k \to ^ ^ ^ ^ ^ ^ L3 : r = 2 i + 8 j + 4k + \alpha(3 i + 9 j + 4k) \to For \alpha = 2, r = 8^i + 26^j + 12k ^
Correct Answer: 3