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JEE Main202226 Jul 2022Evening ShiftMathematicsThree Dimensional GeometryActual

The plane passing through the line L : l x - y + 3 1 - l z = 1 , x + 2 y - z = 2 and perpendicular to the plane 3 x + 2 y + z = 6 is 3 x - 8 y + 7 z = 4 . If θ is the acute angle between the line L and the y -axis, then 415 cos 2 θ is equal to ______.

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Step-by-step solution

Given the plane passing through the line L 1   : l   x - y + 3 1 - l z = 1 , L 2 :   x + 2 y - z = 2 and perpendicular to the plane 3 x + 2 y + z = 6 is 3 x - 8 y + 7 z = 4 , So direction ratio of line L 1 will be n → 1 = l i ^ - j ^ + 3 1 - l k ^ And L 2 will be n → 2 = i ^ + 2 j ^ - k ^ Now direction ratio of line = i ^ j ^ k ^ l - 1 3 1 - l 1 2 - 1 = 6 l - 5 i ^ + 3 - 2 l j ^ + 2 l + 1 k ^ Also plane 3 x - 8 y + 7 z = 4 will contain the line 6 l - 5 i ^ + 3 - 2 l j ^ + 2 l + 1 k ^ So,

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