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JEE Main202331 Jan 2023Morning ShiftMathematicsThree Dimensional GeometryActual

Let the line L : x - 1 2 = y + 1 - 1 = z - 3 1 intersect the plane 2 x + y + 3 z = 16 at the point P . Let the point Q be the foot of perpendicular from the point R 1 , - 1 , - 3 on the line L . If α is the area of triangle P Q R . then α 2 is equal to _____ .

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Step-by-step solution

We have, L : x - 1 2 = y + 1 - 1 = z - 3 1 = λ Any point on L is P 2 λ + 1 , - λ - 1 , λ + 3 It lies on plane 2 x + y + 3 z = 16 , so ⇒ 2 2 λ + 1 + - λ - 1 + 3 λ + 3 = 16 ⇒ 6 λ + 10 = 16 ⇒ λ = 1 ∴ P ≡ 3 , - 2 , 4 Let Q 2 μ + 1 , - μ - 1 , μ + 3 Direction ratios of Q R is 2 μ , - μ , μ + 6 Direction ratios of L is 2 , - 1 , 1 . Since, Q R ⊥ L , so 4 μ + μ + μ + 6 = 0 ⇒ μ = - 1 Q &#

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