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NTA Abhyas JEE Main2020MathematicsThree Dimensional GeometryPractice

Given four points A 2 , 1 , 0 ,   B 1 , 2 , 3 ,   C 3 , 1 , 1 and D 0 , 1 , 3 . Point D lies on the line L which is perpendicular to the plane determined by the points A , B , C . If the point of intersection of the plane determined by the points A , B , C and the line L is a , b , c , then 9 a + b + c is equal to

Correct answer

46

Step-by-step solution

Equation of the plane through A , B , C is x - 2 y - 1 z - 0 1 - 2 2 - 1 3 - 0 3 - 2 1 - 1 1 - 0 = 0 ⇒ x - 2 y - 1 z - 1 1 3 1 0 1 = 0 ⇒ x - 2 1 - y - 1 - 4 + z - 1 = 0 ⇒ x - 2 + 4 y - 4 - z = 0 ⇒ x + 4 y - z - 6 = 0 Equation of the line through D and perpendicular to the plane is x - 0 1 = y - 1 4 = z - 3 - 1 ∵ Point a , b , c is the point of intersection of the line and plane, any general point on the line is λ , 4 λ + 1 , - λ + 3 It must satisfy the equation of the plane, so, λ + 16 λ + 4 + λ - 3 - 6 = 0 ⇒ 18 λ

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