A(2,6,2), B(-4,0, λ), C(2,3,-1) and D(4,5,0),|λ| ≤ 5 are the vertices of a quadrilateral A B C D. If its area is 18 square units, then 5-6 λ is equal to __________.

Solution

Maths sir · 4 steps

Use the diagonals: area of a quadrilateral is half the size of AC⃗ × BD⃗.

1. Diagonal vectors. AC⃗ = C - A = (0, -3, -3) and BD⃗ = D - B = (8, 5, -λ). For any plane quadrilateral, area = 1/2|AC⃗ × BD⃗|.

   AC⃗ × BD⃗ = |î, ĵ, k̂; 0, -3, -3; 8, 5, -λ| = (3λ + 15) î - 24 ĵ + 24 k̂

2. Use the area. Area 18 means |AC⃗ × BD⃗| = 36. So (3λ + 15)² + 576 + 576 = 1296, which gives 9(λ + 5)² = 144, that is (λ + 5)² = 16.
3. Pick the root. λ + 5 = ± 4 gives λ = -1 or λ = -9. Only λ = -1 satisfies |λ| ≤ 5.
4. So. 5 - 6λ = 5 - 6(-1) = 5 + 6 = 11.

The common slip: 59. Ignoring |λ| ≤ 5 and taking λ = -9 gives 5 + 54 = 59, not 11.

The idea to keep. For a quadrilateral, you do not need to split it into triangles: area is half the magnitude of the cross product of its two diagonals. Then let the given condition choose the root.

Answer from the NTA final answer key. Solution by PrepWiser.