Let P(1, 2, 3) be a fixed point and L be the line x-1 2 = y+1 1 = z-2 2 . A point Q moves along the line L . Let R be a point that divides the line segment PQ internally in the ratio 1:2 . If the locus of R is a line L' , and d is the perpendicular distance from the point P to the line L' , then the value of 81d² is equal to

Step-by-step solution

Let the coordinates of the moving point Q on the line L be (2t+1, t-1, 2t+2) . The point R divides the line segment PQ internally in the ratio 1:2 . Using the section formula, the coordinates of R are: R = ( 1(2t+1) + 2(1) 1+2 , 1(t-1) + 2(2) 1+2 , 1(2t+2) + 2(3) 1+2 ) R = ( 2t+3 3 , t+3 3 , 2t+8 3 ) Let R = (x, y, z) . Then x = 2t 3 + 1 , y = t 3 + 1 , z = 2t 3 + 8 3 . This represents the locus of R , which is the line L' . The line L' passes through the point A(1, 1, 8 3 ) and is parallel to the vector b = 2 i +