Three Dimensional Geometry · Question 50

Let L1 : x-11 = y-2-1 = z-12 and L2 : x+1-1 = y-22 = z1 be two lines. Let L3 be a line passing through the point ( , , ) and be perpendicular to both L1 and L2. If L3 intersects L1, then |5 - 11 - 8 | equals :

1. 25
2. 20
3. 16
4. 18

Answer

(1) 25

Solution

Direction of L3 = b1 b2, where b1 = (1, -1, 2), b2 = (-1, 2, 1)

b1 b2 = vmatrix i & j & k \ 1 & -1 & 2 \ -1 & 2 & 1 vmatrix = -5 i - 3 j + k

L3 meets L1 at (1 + t, 2 - t, 1 + 2t), so every point of L3 is

( , , ) = (1 + t - 5s, 2 - t - 3s, 1 + 2t + s)

5 - 11 - 8 = 5(1 + t - 5s) - 11(2 - t - 3s) - 8(1 + 2t + s)

= (5 - 22 - 8) + (5 + 11 - 16)t + (-25 + 33 - 8)s

= -25

|5 - 11 - 8 | = 25

Answer: (1)