Let P be the image of the point Q (4, , 5) with 0 in the line x-1 2 = y-2 1 = z-3 -2 . If R is the point (1,2,3) and the area of the triangle PQR is 10 , then the value of is :

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Let P be the image of the point Q (4, , 5) with > 0 in the line x-1 2 = y-2 1 = z-3 -2 . If R is the point (1,2,3) and the area of the triangle PQR is 10 , then the value of is :

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Correct answer: C. 6

The line L is given by x-1 2 = y-2 1 = z-3 -2 . The point R(1,2,3) lies on the line L . Since P is the image of Q in L , the line L is the perpendicular bisector of the segment PQ . Let S be the foot of the perpendicular from Q(4, , 5) to L . Then the area of PQR is given by 1 2 PQ RS = QS RS . The vector RQ = (4-1) i + ( -2) j + (5-3) k = 3 i + ( -2) j + 2 k . The direction vector of L is d = 2 i + j - 2 k . The length RS is the magnitude of the projection of RQ on d : RS = | RQ d | | d | = |3(2) + ( -2)(1) + 2(-2

Let P be the image of the point Q (4, , 5) with > 0 in the line x-1 2 = y-2 1 = z-3 -2 . If R is the point (1,2,3) and the area of the triangle PQR is 10 , then the value of is :

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