If the shortest distance between the lines (x-1)/2=(y-2)/3=(z-3)/4 and x/1=y/α=(z-5)/1 is 5/√6, then the sum of all possible values of α is

Solution

Maths sir · 4 steps

Write the shortest distance formula with α in it, square both sides, and read the sum off the quadratic.

1. Points and directions. Line 1 passes through A(1, 2, 3) with direction b⃗₁ = (2, 3, 4). Line 2 passes through B(0, 0, 5) with direction b⃗₂ = (1, α, 1). So AB⃗ = (-1, -2, 2).
2. Cross product. b⃗₁ × b⃗₂ = (3 - 4α, 2, 2α - 3). Its squared length is 20α² - 36α + 22. Dotting it with AB⃗ gives -(3 - 4α) - 4 + 2(2α - 3) = 8α - 13.

   (|8α - 13|)/√(20α² - 36α + 22) = 5/√6

3. Square and simplify. 6(64α² - 208α + 169) = 25(20α² - 36α + 22) becomes 116α² + 348α - 464 = 0. Dividing by 116 gives α² + 3α - 4 = 0.
4. So. The roots are α = 1 and α = -4, and both satisfy the original equation. Their sum is 1 + (-4) = -3, option (c).

The idea to keep. Shortest distance is (|AB⃗ · (b⃗₁ × b⃗₂)|)/(|b⃗₁ × b⃗₂|). With an unknown inside, square to get a quadratic and use sum of roots = -b/a.

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