JEE Main 2022 · 26 July · morning · Single correct · +4 / −1

If the system of linear equations.
8x + y + 4z = - 2
x + y + z = 0
λ x - 3y = μ
has infinitely many solutions, then the distance of the point ( λ ,μ , - 1/2 ) from the plane 8x + y + 4z + 2 = 0 is :

  1. (a) 3√5
  2. (b) 4
  3. (c) 26/9
  4. (d) 10/3
Answer and solution

Answer (d) 10/3

Solution

Maths sir · 4 steps

Infinitely many solutions means the third equation must be a mix of the first two. Find that mix.

  1. Determinant is zero. Infinitely many solutions needs the coefficient determinant to vanish. Expanding along the first row gives 8(0+3) - 1(0-λ) + 4(-3-λ) = 12 - 3λ, so λ = 4.
  2. Match the third equation. Write 4x - 3y + 0z = μ as a(8x + y + 4z) + b(x + y + z). The x and z coefficients give 8a + b = 4 and 4a + b = 0, so a = 1, b = -4. The y coefficient checks: 1 - 4 = -3.
  3. Find mu. The right sides must combine the same way: μ = 1 × (-2) + (-4) × 0 = -2. Any other μ makes the system inconsistent. The point is (4, -2, -1/2).

    d = (|8(4) + (-2) + 4(-1/2) + 2|)/√(8² + 1² + 4²) = 30/√81 = 30/9

  4. So. The distance is 10/3, option (d).

The idea to keep. A zero determinant only fixes the coefficients. For infinitely many solutions, the constants must follow the same combination too. That condition gives the second unknown.

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

Study Determinants

Asked about 190 times in JEE Main since 2020. This question is from 2022.