If the shortest distance between the line joining the points $(1,2,3)$ and $(2,3,4)$, and the line $\frac{\mathrm{x}-1}{2}=\frac{\mathrm{y}+1}{-1}=\frac{\mathrm{z}-2}{0}$ is $\alpha$, then $28 \alpha^{2}$ is equal to ______.

JEE Main · 2023 · 25 Jan 2023 · Shift 2 · Three Dimensional Geometry

If the shortest distance between the line joining the points

(1, 2, 3)

and

(2, 3, 4)

, and the line

\frac{x - 1}{2} = \frac{y + 1}{- 1} = \frac{z - 2}{0}

α

, then

28 α^{2}

is equal to ______.

Answer: 18

Solution

The first line runs through

(1, 2, 3)

with direction

(1, 1, 1)

, the second through

(1, - 1, 2)

with direction

(2, - 1, 0)

. Their cross product is

(1, 2, - 3)

, of magnitude

14

, and the join of the base points is

(0, - 3, - 1)

. The shortest distance is

α = \frac{∣ ( 0 , - 3 , - 1 ) \cdot ( 1 , 2 , - 3 ) ∣}{14} = \frac{∣ - 6 + 3∣}{14} = \frac{3}{14}

. Hence

28 α^{2} = 28 \cdot \frac{9}{14} = 18

JEE Main · 2023 · 25 Jan 2023 · Shift 2 · Integer answer

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