JEE Main 2020 · 3 September · morning · Single correct · +4 / −1

The lines
r⃗ = ( î - ĵ ) + l( 2î + k̂ ) and
r⃗ = ( 2î - ĵ ) + m( î + ĵ + k̂ )

  1. (a) do not intersect for any values of l and m
  2. (b) intersect for all values of l and m
  3. (c) intersect when l = 2 and m = 1/2
  4. (d) intersect when l = 1 and m = 2
Answer and solution

Answer (a) do not intersect for any values of l and m

Solution

Maths sir · 4 steps

Write a general point on each line, set them equal, and see if one pair of l, m fits all three components.

  1. General points. A point on the first line is (1 + 2l, -1, l). A point on the second line is (2 + m, -1 + m, m).

    1 + 2l = 2 + m, -1 = -1 + m, l = m

  2. Solve and check. The y-equation gives m = 0, and the z-equation then gives l = 0. Put these into the x-equation: the left side is 1 and the right side is 2. They never match, so no common point exists.
  3. Not parallel either. The directions 2î + k̂ and î + ĵ + k̂ are not multiples of each other. The lines are not parallel and do not meet, so they are skew lines.
  4. So. The lines do not intersect for any values of l and m, which is option (a).

The idea to keep. To test whether two lines meet, solve two component equations for the parameters, then check the third. If the third fails, the lines do not intersect.

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

Study Three Dimensional Geometry

Asked about 300 times in JEE Main since 2020. This question is from 2020.