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JEE Main202229 Jun 2022Morning ShiftMathematicsThree Dimensional GeometryActual

Let P 1 : r → · 2 i ^ + j ^ - 3 k ^ = 4 be a plane. Let P 2 be another plane which passes through the points 2 , - 3 , 2 2 , - 2 , - 3 and 1 , - 4 , 2 . If the direction ratios of the line of intersection of P 1 and P 2 be 16 , α , β , then the value of α + β is equal to ______.

Correct answer

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Step-by-step solution

Given P 1 : r → · 2 i ^ + j ^ - 3 k ^ = 4 i.e. P 1 : 2 x + y - 3 z = 4 Equation of P 2 is given by x - 2 y + 3 z - 2 0 1 - 5 - 1 - 1 0 = 0 ⇒ - 5 x + 5 y + z + 23 = 0 Now n 1 → × n 2 → = i ^ j ^ k ^ 2 1 - 3 - 5 5 1 = 16 i ^ + 13 j ^ + 15 k ^ Let a , b , c be the D R ' s of the line of intersection Then a = 16 λ 15 ; b = 13 λ 15 ; c = 15 λ 15 ∴ α = 13 : β = 15 Hence α + β = 28

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