3D Geometry
Three Dimensional Geometry
nta_pyq_2025_jan
Grade 12
Question:
Let L : y-1 y z+4 1 x-1 3 = -1 = z+1 0 and L : 2 x-2 2 = 0 = \alpha ,\alpha \in R , be two lines, which intersect at the point B. If P is the foot of perpendicular from the point A(1, 1, -1) on L , then the value of 26\alpha( PB) is _________ 2 2
Step-by-Step Solution
Key Concept: Apply the core result for lines and planes in three dimensions and simplify using the given constraints.
Point B (216) (3\lambda + 1, -\lambda + 1, -1) ≡ (2\mu + 2, 0, \alpha\mu - 4) 3\lambda + 1 = 2\mu + 2 - \lambda + 1 = 0 - 1 = \alpha\mu - 4 \lambda = 1, \mu = 1, \alpha = 3 B(4, 0, -1) Let Point ' P ' is (2\delta + 2, 0, 3\delta - 4) Dr's of AP < 2\delta + 1, -1, 3\delta - 3 > 7 AP \perp L2 \Rightarrow \delta = 13 40 -31 P ( , 0, ) 13 13 2 144 324 \therefore 26\alpha(P B) = 26 \times 3 \times ( + ) 169 169 = 216
Correct Answer: 216