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JEE Main20231 Feb 2023Morning ShiftMathematicsVector AlgebraActual

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 _____ .

Correct answer

0

Step-by-step solution

Given: A 2 , 6 , 2 , B - 4 , 0 , λ , C 2 , 3 , - 1 D 4 , 5 , 0 Area of quadrilateral = 1 2 A C → × B D → = 18 ⇒ A C → × B D → 36 Now, A C → × B D → = i ^ j ^ k ^ 0 - 3 - 3 8 5 - λ ⇒ A C → × B D → = 3 λ + 15 i ^ - 0 + 24 j ^ + 0 + 24 k ^ ⇒ A C → × B D → = 3 λ + 15 i ^ - 24 j ^ + 24 k ^ So, A C → × B D → = 36 ⇒ 3 λ + 15 2 + 24 2 + 24 2 = 36 ⇒ 9 λ 2 + 225 + 9

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