3D Geometry
Lines and direction cosines
Grade 12
Question:
<p>\(ABC\) is a triangle in a plane with vertices \(A(2, 3, 5)\), \(B(-1, 3, 2)\) and \(C(\lambda, 5, \mu)\). If the median through \(A\) is equally inclined to the coordinate axes, then the value of \((\lambda^3 + \mu^3 + 5)\) is</p>
<p>1130</p>
<p>1348</p>
<p>1077</p>
<p>676</p>
Step-by-Step Solution
Key Concept: A line equally inclined to coordinate axes has direction ratios (1,1,1), meaning the direction cosines satisfy l=m=n=1/√3. Use this constraint on the median from A to the midpoint of BC.
Step 1: Find the midpoint M of BC. M = ((-1+λ)/2, (3+5)/2, (2+μ)/2) = ((-1+λ)/2, 4, (2+μ)/2) Step 2: Find the direction ratios of median AM from A(2,3,5) to M. Direction ratios = ((-1+λ)/2 - 2, 4-3, (2+μ)/2 - 5) = ((λ-5)/2, 1, (μ-8)/2) Step 3: For a line equally inclined to coordinate axes, direction ratios must be proportional to (1,1,1). This means: (λ-5)/2 : 1 : (μ-8)/2 = 1 : 1 : 1 Step 4: From the ratio equality: (λ-5)/2 = 1 ⟹ λ-5 = 2 ⟹ λ = 7 (μ-8)/2 = 1 ⟹ μ-8 = 2 ⟹ μ = 10 Step 5: Calculate λ^3 + μ^3 + 5. λ^3 + μ^3 + 5 = 7^3 + 10^3 + 5 = 343 + 1000 + 5 = 1348 ∴ Answer: B
Correct Answer: B