If the lines $\frac{x-1}{2}=\frac{2-y}{-3}=\frac{z-3}{\alpha}$ and $\frac{x-4}{5}=\frac{y-1}{2}=\frac{z}{\beta}$ intersect, then the magnitude of the minimum value of $8 \alpha \beta$ is

JEE Main · 2023 · 06 Apr 2023 · Shift 2 · Three Dimensional Geometry

If the lines

\frac{x - 1}{2} = \frac{2 - y}{- 3} = \frac{z - 3}{α}

and

\frac{x - 4}{5} = \frac{y - 1}{2} = \frac{z}{β}

intersect, then the magnitude of the minimum value of

8 α β

Answer: 18

Solution

For the lines to intersect, the vectors along the lines and the vector joining points on them must be coplanar. The first line passes through

(1, 2, 3)

with direction

(2, 3, α)

; the second through

(4, 1, 0)

with direction

(5, 2, β)

. The vector joining the points is

(- 3, 1, 3)

. The coplanarity condition gives

25 - 3 321 α β 3 = 0

, which simplifies to

α - β = 3

. Substituting

α = 3 + β

into

8 α β = 8 β (3 + β) = 8 (β + \frac{3}{2})^{2} - 18

, the minimum value has magnitude 18.

JEE Main · 2023 · 06 Apr 2023 · Shift 2 · Integer answer

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