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Let Q be the foot of the perpendicular drawn from the point R(4, 1, ) to the line x-1 2 = y+1 3 = z-2 -1 . If P(1, 1, 1) is a point such that the length of the line segment PQ is 5 , then the sum of the squares of all possible values of is equal to ________

Correct answer

196

Step-by-step solution

Let the coordinates of the general point on the given line be Q(2 +1, 3 -1, - +2) . Given that the distance PQ = 5 , we have: PQ^2 = (2 +1-1)^2 + (3 -1-1)^2 + (- +2-1)^2 = 5 (2 )^2 + (3 -2)^2 + (1- )^2 = 5 4 ^2 + 9 ^2 - 12 + 4 + 1 - 2 + ^2 = 5 14 ^2 - 14 + 5 = 5 14 ( - 1) = 0 Thus, = 0 or = 1 . The direction ratios of the line segment RQ are (2 +1-4, 3 -1-1, - +2- ) = (2 -3, 3 -2, 2- - ) . Since RQ is perpendicular to the given line, the dot product of their direction ratios is zero: 2(2 -3) + 3(3 -2) - 1(2- - ) =

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