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A line x - a 2 = y - b 3 = z - c 4 intersects a plane x - y + z = 4 at a point where the line x - 1 2 = y + 3 5 = z + 1 2 meets the plane. Also, a plane a x - 2 y + b z = 3 meet them at the same point, then 11 a + b + c is equal to

Correct answer

210

Step-by-step solution

Any general point on the line x - 1 2 = y + 3 5 = z + 1 2 is 2 λ + 1,5 λ - 3,2 λ - 1 It must satisfy x - y + z = 4 ⇒ 2 λ + 1 - 5 λ + 3 + 2 λ - 1 = 4 ⇒ - λ + 3 = 4 ⇒ λ = - 1 So, the point of intersection of the plane and the line is - 1 , - 8 , - 3 This point also lies on a x - 2 y + b z = 3 ⇒ - a + 16 - 3 b = 3 ⇒ a + 3 b = 13 … . i This point also satisfy x - a 2 = y - b 3 = z - c 4 ⇒ 1 + a 2 = 8 + b 3 = 3 + c 4 … i i By taking

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