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JEE MainMathematicsThree Dimensional Geometry

Line L₁ passes through the point (a, 2, 3) and is parallel to the vector i + 2 j + 3 k . Line L₂ passes through the point (4, b, 1) and is parallel to the vector 2 i + 3 j + 4 k . If L₁ and L₂ intersect, then the minimum value of 5(a^2 + b^2) is ____.

Correct answer

36

Step-by-step solution

Let the position vectors of the points on L₁ and L₂ be a₁ = a i + 2 j + 3 k and a₂ = 4 i + b j + k . The lines are parallel to b₁ = i + 2 j + 3 k and b₂ = 2 i + 3 j + 4 k respectively. Since the two lines intersect, they must be coplanar. Thus, the vector a₂ - a₁ must be coplanar with b₁ and b₂ . a₂ - a₁ = (4-a) i + (b-2) j - 2 k The condition for coplanarity is the scalar triple product being zero: vmatrix 4-a & b-2 & -2 1 & 2 & 3 2 & 3 & 4 vmatrix = 0 Expanding the determinant: (4-a)(8 - 9) - (b-2)(4 - 6) - 2(3 -

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