JEE Main · 2023 · 01 Feb 2023 · Shift 1 · Three Dimensional Geometry

The shortest distance between the lines

\frac{x - 5}{1} = \frac{y - 2}{2} = \frac{z - 4}{- 3}

and

\frac{x + 3}{1} = \frac{y + 5}{4} = \frac{z - 1}{- 5}

(A) $73$
(B) $53$
(C) $63$ ✓ Correct
(D) $43$

Answer: (C)

Solution

Taking

a = (1, 2, - 3)

and

b = (1, 4, - 5)

as the direction vectors,

a \times b = (2, 2, 2)

with magnitude

23

. The vector joining the points

(5, 2, 4)

and

(- 3, - 5, 1)

(8, 7, 3)

, whose projection onto

a \times b

has magnitude

∣16 + 14 + 6∣ = 36

. The shortest distance is

\frac{36}{2 3} = 63

JEE Main · 2023 · 01 Feb 2023 · Shift 1 · Single correct option

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