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Question:

Three lines are given by r→ = λi^, λ∈R; r→ = μ(i^ + j^), μ∈R; and r→ = v(i^ + j^ + k^), v∈R. Let the lines cut the plane x + y + z = 1 at the points A, B and C respectively. If the area of the triangle ABC is Δ, then the value of (6Δ)² equals _________.

0.75

0.1

0.85

0.65

Solution:

Correct option is A. 0.75
Put (λ,0,0) in x+y+z=1 ⇒ λ=1 ⇒ P(1,0,0)
Put (μ,μ,0) ⇒ 2μ=1 ⇒ Q(1/2,1/2,0)
Put (γ,γ,γ) ⇒ γ=1/3 ⇒ R(1/3,1/3,1/3)
Area of triangle PQR = 1/2|PQ→ × PR→| = 1/2|(i^ - j^/2) × (2i^ - j^ - k^/3)| = 1/12|i^ + j^ + k^| = √3/12.
(6Δ)² = (6√3/12)² = (√3/2)² = 3/4 = 0.75