Let the line L intersect the lines x-2=-y=z-1,2(x+1)=2(y-1)=z+1 and be parallel to the line (x-2)/3=(y-1)/1=(z-2)/2. Then which of the following points lies on L ?

Solution

Maths sir · 5 steps

Take a general point on each given line; the segment joining them must point along (3, 1, 2).

1. General points. On the first line put x-2=-y=z-1=s, giving P(2+s, -s, 1+s). On the second put 2(x+1)=2(y-1)=z+1=2a, giving Q(-1+a, 1+a, -1+2a).
2. Parallel condition. L passes through P and Q and is parallel to (3, 1, 2), so PQ⃗ = λ(3, 1, 2) for some λ.

   PQ⃗ = (a - s - 3, a + s + 1, 2a - s - 2) = λ(3, 1, 2)

3. Solve. Subtracting the first component equation from the second gives 2s + 4 = -2λ, so λ = -2 - s and then a = -3 - 2s. The third equation becomes -8 - 5s = -4 - 2s, so s = -4/3, a = -1/3, λ = -2/3.
4. Equation of L. With s = -4/3, P = (2/3, 4/3, -1/3). So L is (2/3 + 3k, 4/3 + k, -1/3 + 2k). For x = -1/3 we need k = -1/3, which gives y = 1 and z = -1.
5. So. The point of L with x = -1/3 is (-1/3, 1, -1), which is option (a).

The idea to keep. For a line that meets two given lines and has a known direction, write a general point on each line and make the joining vector proportional to that direction.

Answer from the NTA final answer key. Solution by PrepWiser.