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JEE MainMathematicsThree Dimensional Geometry

Let A(1, -1, 2) be a vertex of a rhombus ABCD . The diagonal BD lies on the line x-2 1 = y-1 -2 = z+1 2 . If the area of the rhombus is 18 5 sq. units, and the coordinates of B and D are (x₁, y₁, z₁) and (x₂, y₂, z₂) respectively, then the value of (x₁^2 + y₁^2 + z₁^2) + (x₂^2 + y₂^2 + z₂^2) is equal to

Correct answer

184

Step-by-step solution

Let M be the point of intersection of the diagonals of the rhombus. M is the foot of the perpendicular from A(1, -1, 2) to the line containing BD . Any point on the line is given by M(r+2, -2r+1, 2r-1) . The direction ratios of AM are (r+1, -2r+2, 2r-3) . Since AM is perpendicular to the line, its dot product with the line's direction ratios (1, -2, 2) is zero: 1(r+1) - 2(-2r+2) + 2(2r-3) = 0 r + 1 + 4r - 4 + 4r - 6 = 0 9r = 9 r = 1 Substituting r = 1 , the coordinates of M are (3, -1, 1) . The length of AM is (3-1

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