3D Geometry
Three Dimensional Geometry
nta_pyq_2025_jan
Grade 12
Question:
Let A(x, y, z) be a point in xy-plane, which is equidistant from three points (0, 3, 2), (2, 0, 3) and ( 0, 0, 1 ). Let B = (1, 4, -1) and C = (2, 0, -2). Then among the statements (S1) : △ABC is an isosceles right angled triangle, and 9\sqrt2 (S2) : the area of △ABC is 2 ,
both are true
only (S2) is true
only (S1) is true
both are false
Step-by-Step Solution
Key Concept: Apply the core result for lines and planes in three dimensions and simplify using the given constraints.
A(x, y, z) Let P(0, 3, 2), Q(2, 0, 3), R(0, 0, 1) (3) AP = AQ = AR 2 2 2 2 2 2 2 x + (y - 3) + (z - 2) = (x - 2) + y + (z - 3) = x + 2 2 y + (z - 1) In xy plane z = 0 So, x - 4x + 4 + y + 9 = x + y + 1 2 2 2 2 x = 3 2 2 2 9 + y - 6y + 9 + 4 = x + y + 1 So, A(3, 2, 0) also B(1, 4, -1)&C(2, 0, -2) Now AB = \sqrt4 + 4 + 1 = 3 AC = \sqrt1 + 4 + 4 = 3 BC = \sqrt1 + 16 + 1 = \sqrt18 AB = AC isosceles $\Delta$&AB + AC = BC 2 2 2 right angle $\Delta$ Area of △ABC = 1 2 \times base.height 1 9 \times 3 \times 3 = 2 2 So only S is true 1
Correct Answer: 3