A straight line passes through the point A(1, 2, 3) and intersects the lines x-4 1 = y-2 -1 = z-2 1 and x+5 2 = y+1 1 = z+1 3 at the points P and Q , respectively. The square of the length of the line segment PQ is equal to:

Step-by-step solution

Let the coordinates of point P on the first line be (4+ , 2- , 2+ ) . Let the coordinates of point Q on the second line be (-5+2 , -1+ , -1+3 ) . Since the line passes through A(1, 2, 3) , the points A , P , and Q are collinear. Thus, the vectors AP and AQ must be proportional. AP = (3+ , - , -1+ ) AQ = (-6+2 , -3+ , -4+3 ) Using the condition of proportionality: 3+ -6+2 = - -3+ = -1+ -4+3 From the first two ratios: (3+ )(-3+ ) = - (-6+2 ) -9 + 3 - 3 + = 6 - 2 3 - 9 + 3 - 9 = 0 - 3 + - 3 = 0 ( +1)( -3) = 0 So, eith