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NDA 1 2024 Maths, question 57

Topic: Analytical geometry

Q57

If the direction cosines ⟨l,m,n⟩⟨l,m,n⟩ of a line are connected by relation l+2m+n=0l+2m+n=0, 2l−2m+3n=02l-2m+3n=0, then what is the value of l2+m2−n2l^{2}+m^{2}-n^{2} ?

  1. (a) 1101\frac{1}{101}
  2. (b) 29101\frac{29}{101}Correct answer
  3. (c) 41101\frac{41}{101}
  4. (d) 92101\frac{92}{101}

Solution.

Answer: (b) 29101\frac{29}{101}

  1. The line of intersection has direction equal to the cross product of the plane normals: (1, 2, 1) × (2, −2, 3) = (8, −1, −6).
  2. Its squared length is 101.
  3. Therefore the required combination of squared direction cosines is (64 + 1 − 36)/101 = 29/101.