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JEE Main202118 Mar 2021Morning ShiftMathematicsThree Dimensional GeometryActual

Let the plane a x + b y + c z + d = 0 bisect the line joining the points 4 , - 3 , 1 and 2 , 3 , - 5 at the right angles. If a ,   b ,   c ,   d are integers, then the minimum value of a 2 + b 2 + c 2 + d 2 is

Correct answer

28

Step-by-step solution

Mid-point of 4 , - 3 , 1 and 2 , 3 , - 5 is 3 , 0 , - 2 which lies on the plane. Direction ratios of normal vector to the plane are 1 , - 3 , 3 Equation of Plane is 1 ( x - 3 ) - 3 ( y - 0 ) + 3 ( z + 2 ) = 0 x - 3 y + 3 z + 3 = 0 a 2 + b 2 + c 2 + d 2 min = 28

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