If the shortest distance between the lines
r⃗ = ( - î + 3k̂ ) + λ ( î - aĵ )
and r⃗ = ( - ĵ + 2k̂ ) + μ ( î - ĵ + k̂ ) is √(2/3), then the integral value of a is equal to ___________.

Solution

Maths sir · 5 steps

Use the shortest distance formula, square it to get a quadratic in a, and keep the integer root.

1. Read off vectors. From the first line, the point is a⃗₁ = (-1, 0, 3) and the direction is b⃗₁ = (1, -a, 0). From the second line, the point is a⃗₂ = (0, -1, 2) and the direction is b⃗₂ = (1, -1, 1).

   d = (|(a⃗₂ - a⃗₁)·(b⃗₁ × b⃗₂)|)/(|b⃗₁ × b⃗₂|)

2. Compute the pieces. a⃗₂ - a⃗₁ = (1, -1, -1) and b⃗₁ × b⃗₂ = (-a, -1, a - 1). Their dot product is -a + 1 - (a - 1) = 2(1 - a). The magnitude is √(a² + 1 + (a-1)²) = √(2a² - 2a + 2).
3. Set up the equation. Put d = √(2/3) and square both sides: 4(1-a)²/2(a² - a + 1) = 2/3. This gives 3(1-a)² = a² - a + 1, which simplifies to 2a² - 5a + 2 = 0.
4. Solve. (2a - 1)(a - 2) = 0, so a = 2 or a = 1/2. Check a = 2: d = 2/√6 = √(2/3), which matches.
5. So. The integral value is a = 2.

The common slip: 0.5. Keeping the other root of 2a² - 5a + 2 = 0 gives 0.5. That root is valid for the distance, but the question asks for the integral value.

The idea to keep. Shortest distance between skew lines is the projection of the joining vector onto b⃗₁ × b⃗₂. Squaring removes the modulus, so check that each root is allowed by the question.

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