If Q(1,0,1) is the image of the point P(a, b, c) in the line x+1 2 = y-3 -2 = z -1 then a+b+c is equal to

Step-by-step solution

Let the line be L: x+1 2 = y-3 -2 = z -1 = k . Any point M on the line is given by M(2k-1, -2k+3, -k) . Since Q(1, 0, 1) is the image of P(a, b, c) in the line, M is the midpoint of PQ . Thus, M = ( a+1 2 , b+0 2 , c+1 2 ) . Equating the coordinates of M with the parametric form: a+1 2 = 2k-1 a = 4k-3 b 2 = -2k+3 b = -4k+6 c+1 2 = -k c = -2k-1 The vector PQ = (1-a, 0-b, 1-c) = (1-(4k-3), -( -4k+6), 1-(-2k-1)) = (4-4k, 4k-6, 2k+2) must be perpendicular to the direction vector of the line v = (2, -2, -1) . Therefore,