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JEE Main202330 Jan 2023Morning ShiftMathematicsThree Dimensional GeometryActual

If λ 1 < λ 2 are two values of λ such that the angle between the planes P 1 : r → ( 3 i ^ - 5 j ^ + k ^ ) = 7 and P 2 : r → · ( λ i ^ + j ^ - 3 k ^ ) = 9 is sin - 1 2 6 5 , then the square of the length of perpendicular from the point 38 λ 1 , 10 λ 2 , 2 to the plane P 1 is

Correct answer

315

Step-by-step solution

Given: P 1 : r → · 3 i ^ - 5 j ^ + k ^ = 7 So, normal vector is n → 1 = 3 i ^ - 5 j ^ + k ^ P 2 : r → · λ i ^ + j ^ - 3 k ^ = 9 So, normal vector is n → 2 = λ i ^ + j ^ - 3 k ^ Now, angle between planes P 1 and P 2 is θ = sin - 1 2 6 5 ⇒ sin θ = 2 6 5 ⇒ cos θ = 1 5 Also, cos θ = n → 1 · n → 2 n → 1 n → 2 ⇒ 1 5 = 3 i ^ - 5 j ^ + k ^ · λ i ^ + j ^ - 3 k ^ 35 λ 2 + 10 ⇒ 1 5 = 3 λ - 8

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