If the lines (x-1)/2=(2-y)/-3=(z-3)/α and (x-4)/5=(y-1)/2=z/β intersect, then the magnitude of the minimum value of 8 α β is _____________.

Solution

Maths sir · 4 steps

Put a general point on each line and match x and y first. The z-coordinate then ties α to β.

1. Fix the first line. First rewrite (2-y)/-3 as (y-2)/3. The first line then passes through (1, 2, 3) with direction (2, 3, α). The second passes through (4, 1, 0) with direction (5, 2, β).
2. Match x and y. General points are (1+2t, 2+3t, 3+α t) and (4+5s, 1+2s, β s). Equating x and y gives 2t-5s=3 and 3t-2s=-1, which solve to t=-1, s=-1.
3. Use z. At the common point the z-coordinates must also agree: 3-α = -β. So β = α - 3.

   8αβ = 8α(α-3) = 8(α-3/2)² - 18

4. So. The square is never negative, so the minimum value of 8αβ is -18, reached at α = 3/2. Its magnitude is 18.

The common slip: -18. Stopping at the minimum value itself gives -18. The question asks for its magnitude, which is 18.

The idea to keep. For intersecting lines, equate general points. Use two coordinates to find the parameters, and the third gives the condition. Then complete the square to minimise.

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