3D Geometry
Three Dimensional Geometry
nta_pyq_2025_jan
Grade 12
Question:
Let the point A divide the line segment joining the points P (-1, -1, 2) and Q(5, 5, 10) internally in the ratio - -\to - -\to - -\to - -\to . If O is the origin and (OQ ⋅ OA) - , then the value of r is : 1 2 r : 1(r > 0) |OP \times OA| = 10 5
Step-by-Step Solution
Key Concept: Apply the core result for lines and planes in three dimensions and simplify using the given constraints.
(4) 5r - 1 5r - 1 10r + 2 A = ( , , ) r + 1 r + 1 r + 1 - -\to - -\to 1 --\to - -\to 2 (OQ ⋅ OA) - |OP \times OA| = 10 5 - -\to ^ ^ ^ OQ = 5 i + 5 j + 10k - -\to 5r - 1 5r - 1 10r + 2 ^ ^ ^ OA = i + j + k r + 1 r + 1 r + 1 - -\to ^ ^ ^ OP = - i - j + 2k - -\to - -\to 1 5r - 1 ∣ OP \times OA = ∣ 5r - 1 10r + 2 r + 1 ∣ 5 1 ^ ^ = ( i (20r) - j(20r)) r + 1 5r - 1 5r - 1 10r + 2 = 5( ) + 5( ) + 10 ( ) r + 1 r + 1 r + 1 2 1 2 \times 400r - ( ) = 10 2 5 (r + 1) 2 150r + 10 1 2 \times 400r - ( ) = 10 2 r + 1 5 (r + 1) 2 2 (150r + 10)(r + 1) - 160r = 10(r + 1) 2 2 (15r + 1)(r + 1) - 16r = (r + 1) 2 2 2 15r + 16r + 1 - 16r = r + 2r + 1 2 - 2r + 14r = 0 r = 0, 7
Correct Answer: 4