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JEE Main202225 Jun 2022Morning ShiftMathematicsThree Dimensional GeometryActual

Let the lines L 1 : r → = λ i ^ + 2 j ^ + 3 k ^ , λ ∈ R and L 2 : r → = i ^ + 3 j ^ + k ^ + μ ( i ^ + j ^ + 5 k ^ ) ; μ ∈ R , intersect at the point S . If a plane a x + b y - z + d = 0 passes through S and is parallel to the lines L 1 and L 2 , then the value of a + b + d is equal to ______.

Correct answer

0

Step-by-step solution

L 1 ≡ x = y 2 = z 3 = λ passes through 0 , 0 , 0 and parallel to i ^ + 2 j ^ + 3 k ^ L 2 ≡ x - 1 = y - 3 = z - 1 5 = μ passes through 1 , 3 , 1 and parallel to i ^ + j ^ + 5 k ^ i.e. λ = 1 + μ ,   2 λ = 3 + μ ⇒ μ = 1 Both the lines intersect at 2 , 4 , 6 Normal to the plane will be i ^ j ^ k ^ 1 2 3 1 1 5 = 7 i ^ - 2 j ^ - k ^ Since plane is parallel to both the given lines, the equation of the plane will be 7 x - 2 - 2 y - 4 - z - 6 = 0 ⇒ 7 x - 2 y -

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