Three Dimensional Geometry · Question 51

Let a = i + 2 j + k and b = 2 i + 7 j + 3 k. Let L1 : r = (- i + 2 j + k) + a, R and L2 : r = ( j + k) + b, R be two lines. If the line L3 passes through the point of intersection of L1 and L2 and is parallel to a + b, then L3 passes through the point :

1. (-1, -1, 1)
2. (2, 8, 5)
3. (8, 26, 12)
4. (5, 17, 4)

Answer

(3) (8, 26, 12)

Solution

General point of L1: (-1 + , 2 + 2 , 1 + )

General point of L2: (2 , 1 + 7 , 1 + 3 )

At the point of intersection:

1 + = 1 + 3 = 3

-1 + = 2 -1 + 3 = 2 = 1, = 3

Check y : 2 + 2(3) = 8 = 1 + 7(1) ✓

point of intersection = (2, 8, 4)

a + b = 3 i + 9 j + 4 k

L3 : x - 23 = y - 89 = z - 44 = t

point (2 + 3t, 8 + 9t, 4 + 4t)

For t = 2: (8, 26, 12), which lies on L3.

(For the other options no single value of t satisfies all three coordinates.)

Answer: (3)