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JEE MainMathematicsThree Dimensional Geometry

A line L passing through the point P(2, 3, 4) intersects the lines L₁ : x-1 1 = y+1 2 = z-2 1 and L₂ : x+2 2 = y-1 1 = z+1 3 . If the direction ratios of the line L are a, b, c where a, b, c are coprime integers and a > 0 , then the value of a + 2b + 3c is equal to ________.

Correct answer

11

Step-by-step solution

Let the line L intersect L₁ at M and L₂ at N . The line L lies in the plane ₁ containing the point P(2, 3, 4) and the line L₁ . The line L₁ passes through A₁(1, -1, 2) and has direction vector d₁ = i + 2 j + k . PA₁ = (1-2) i + (-1-3) j + (2-4) k = - i - 4 j - 2 k . The normal vector to plane ₁ is n₁ = PA₁ d₁ = vmatrix i & j & k -1 & -4 & -2 1 & 2 & 1 vmatrix = 0 i - j + 2 k . Similarly, the line L lies in the plane ₂ containing P(2, 3, 4) and L₂ . The line L₂ passes through A₂(-2, 1, -1) and has direction vector d

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