3D Geometry
Three Dimensional Geometry
nta_pyq_2025_jan
Grade 12
Question:
Let P be the image of the point Q(7, -2, 5) in the line L : y+1 x-1 2 = 3 = z 4 and R(5, p, q) be a point on L. Then the square of the area of △P QR is ________.
Step-by-Step Solution
Key Concept: Apply the core result for lines and planes in three dimensions and simplify using the given constraints.
(957) Let R(2\lambda + 1, 3\lambda - 1, 4\lambda) 2\lambda + 1 = 5 \lambda = 2 R(5, 5, 8) let T(2\lambda + 1, 3\lambda - 1, 4\lambda) - -\to ^ ^ ^ QT = (2\lambda - 6) i + (3\lambda + 1) j + (4\lambda - 5)k \to ^ ^ ^ b = 2 i + 3 j + 4k - -\to \to QT ⋅ b = 0 4\lambda - 12 + 9\lambda + 3 + 16\lambda - 20 = 0 \lambda = 1 T(3, 2, 4) QT = \sqrt33 RT = \sqrt29 2 1 2 ( area of $\Delta$PQR) = ( \sqrt29 ⋅ 2\sqrt33) 2 = 957
Correct Answer: 957