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JEE Main20231 Feb 2023Evening ShiftMathematicsThree Dimensional GeometryActual

The point of intersection C of the plane 8 x + y + 2 z = 0 and the line joining the points A - 3 , - 6 , 1 and B 2 , 4 , - 3 divides the line segment A B internally in the ratio k : 1 . If a , b , c ( | a | , | b | , | c | are coprime) are the direction ratios of the perpendicular from the point C on the line 1 - x 1 = y + 4 2 = z + 2 3 , then a + b + c is equal to _____ .

Correct answer

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Step-by-step solution

Given, Plane: 8 x + y + 2 z = 0 Now equation of line A B : x - 2 5 = y - 4 10 = z + 3 - 4 = λ Any point on line A B is given by, 5 λ + 2 , 10 λ + 4 , - 4 λ - 3 Now using perpendicular condition to find the point of intersection of line and plane, 8 5 λ + 2 + 10 λ + 4 - 8 λ - 6 = 0 ⇒ λ = - 1 3 So, point of intersection will be C 1 3 , 2 3 , - 5 3 Now let L : x - 1 - 1 = y + 4 2 = z + 2 3 = μ Now any point on the line L is given by, - μ + 1 , 2 μ - 4 , 3 &#

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