3D Geometry
Three Dimensional Geometry
nta_pyq_2025_jan
Grade 12
Question:
Let a line pass through two distinct points P (-2, -1, 3) and Q, and be parallel to the vector 3^i + 2^j + 2k ^ . If the distance of the point Q from the point R(1, 3, 3) is 5 , then the square of the area of △P QR 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.
Equation of line P Q is: (2) x+2 y+1 z-3 = = = r( say ) 3 2 2 Let coordinate of Q = (3r - 2, 2r - 1, 2r + 3) ∵ PR = 5 Then 2 2 2 (3r - 2 - 1) + (2r - 1 - 3) + (2r + 3 - 3) = 25 \therefore r = 0 or 2 \therefore Coordinate of Q = (4, 3, 7) - -\to - -\to 2 ∣ 1 ∣ \therefore square of area of △P QR = ∣ (P Q \times P R)∣ ∣ 2 ∣ 2 ∣ 1 ∣ ^ ^ ^ ^ ^ = ∣ (6 i + 4 j + 4k) \times (3 i + 4 j)∣ ∣ 2 ∣ ^ ^ ^ 2 = | - 8 i + 6 j + 6k| = 136
Correct Answer: 2