Let P(3, 2, 1) be a point and L be the line x-1 2 = y-2 1 = z-3 2 . Two points A and B lie on the line L such that PA = PB . If the area of the triangle PAB is 12 2 , then the value of PA^2 is ________.

Step-by-step solution

Let T be the foot of the perpendicular from P to the line L . Since PA = PB , the triangle PAB is isosceles and T is the midpoint of AB . Any point on the line L can be written as T(2t+1, t+2, 2t+3) . The direction ratios of the line L are (2, 1, 2) . The direction ratios of the vector PT are (2t+1-3, t+2-2, 2t+3-1) = (2t-2, t, 2t+2) . Since PT is perpendicular to L , their dot product is zero: 2(2t-2) + 1(t) + 2(2t+2) = 0 4t - 4 + t + 4t + 4 = 0 9t = 0 t = 0 Thus, the coordinates of T are (1, 2, 3) . The perpendic