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NTA Abhyas JEE Main2020MathematicsThree Dimensional GeometryPractice

If the planes x - c y - b z = 0 ,   c x - y + a z = 0 and b x + a y - z = 0 pass through a straight line, then the value of a 2 + b 2 + c 2 + 2 a b c is

Correct answer

1

Step-by-step solution

Given planes are: x - c y - b z = 0 ...(i) c x - y + a z = 0 ...(ii) b x + a y - z = 0 ...(iii) Equation of the plane passing through the line of intersection of the plane (i) and (ii) may be taken as: x - c y - b z + λ c x - y + a z = 0 o r x 1 + λ c - y c + λ + z - b + a λ = 0 ...(iv) If plane (iii) and (iv) are the same, then ∴ 1 + c λ b = - ( c + λ ) a = - b + a λ - 1 or λ = - a + b c a c + b and λ = - ( a b + c ) 1 - a 2 ∴ - a + b c a c + b = - a b + c 1 - a 2 ⇒ a - a 3 + b c - a 2 b c = a 2 b c + a c 2 + a b

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