JEE Main · 2023 · 24 Jan 2023 · Shift 2 · Three Dimensional Geometry

The shortest distance between the lines

\frac{x - 2}{3} = \frac{y + 1}{2} = \frac{z - 6}{2}

and

\frac{x - 6}{3} = \frac{1 - y}{2} = \frac{z + 8}{0}

is equal to

Answer: 14

Solution

The first line has point

(2, - 1, 6)

and direction

(3, 2, 2)

; rewritten as

\frac{x - 6}{3} = \frac{y - 1}{- 2} = \frac{z + 8}{0}

, the second has point

(6, 1, - 8)

and direction

(3, - 2, 0)

. Then

b_{1} \times b_{2} = (4, 6, - 12)

with magnitude

196 = 14

, and the join

(4, 2, - 14)

gives

(4, 2, - 14) \cdot (4, 6, - 12) = 196

. The shortest distance is

\frac{∣196∣}{14} = 14.

JEE Main · 2023 · 24 Jan 2023 · Shift 2 · Integer answer

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