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JEE Main202329 Jan 2023Morning ShiftMathematicsThree Dimensional GeometryActual

Let the equation of the plane P containing the line x + 10 = 8 - y 2 = z be a x + b y + 3 z = 2 ( a + b ) and the distance of the plane P from the point ( 1 , 27 , 7 ) be c . Then a 2 + b 2 + c 2 is equal to

Correct answer

0

Step-by-step solution

Given, The line x + 10 1 = y - 8 - 2 = z 1 which passes through the point ( - 10 , 8 , 0 ) Also given the equation of plane a x + b y + 3 z = 2 ( a + b ) and the point ( - 10 , 8 , 0 ) will lie on it as plane contains the line, So, a - 10 + b × 8 + 3 × 0 = 2 ( a + b ) ⇒ b = 2 a   . . . . . . 1 Now product of direction ratio of line and direction ratio of plane will be zero as they both are perpendicular, So, a × 1 + b × - 2 + 3 × 1 = 0 ⇒ a - 2 b + 3 = 0   . . . . . . .

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