3D Geometry
Three Dimensional Geometry
nta_pyq_2025_jan
Grade 12
Question:
If the image of the point (4, 4, 3) in the line x-1 = y-2 = z-1 is (\alpha, \beta, \gamma), then \alpha + \beta + \gamma is equal to 2 1 3
Step-by-Step Solution
Key Concept: Apply the core result for lines and planes in three dimensions and simplify using the given constraints.
(1) Let coordinate of R = (2r + 1, r + 2, 3r + 1) ∵ PR is perpendicular to given line. \therefore (2r - 3) ⋅ 2 + (r - 2) ⋅ 1 + (3r - 2) ⋅ 3 = 0 \therefore r = 1 \therefore Coordinate of R = (3, 3, 4) \therefore (\alpha, \beta, \gamma) = (2, 2, 5) \therefore \alpha + \beta + \gamma = 9
Correct Answer: 1