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JEE Main20238 Apr 2023Evening ShiftMathematicsThree Dimensional GeometryActual

Let P 1 be the plane 3 x - y - 7 z = 11 and P 2 be the plane passing through the points ( 2 , - 1 , 0 ) , ( 2 , 0 , - 1 ) , and ( 5 , 1 , 1 ) . If the foot of the perpendicular drawn from the point ( 7 , 4 , - 1 ) on the line of intersection of the planes P 1 and P 2 is ( α , β , γ ) , then α + β + γ is equal to

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Step-by-step solution

Given, P 1 be the plane 3 x - y - 7 z = 11 , P 2 be the plane passing through the points ( 2 , - 1 , 0 ) , ( 2 , 0 , - 1 ) , and ( 5 , 1 , 1 ) , Now finding the equation of plane P 2 , we get x - 2 y + 1 z 2 - 2 0 - - 1 - 1 - 0 5 - 2 1 - 0 1 - - 1 = 0 ⇒ x - 2 y + 1 z 0 1 - 1 3 1 2 = 0 ⇒ 3 x - 2 + 3 - y - 1 - 3 z = 0 ⇒ x - 2 - y - 1 - z = 0 ⇒ x - y - z = 3 Now direction ratio of line of intersection of planes 3 x - y - 7 z = 11   &   x - y - z = 3 is given by, i ^ j ^ k ^ 3 - 1

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