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

The shortest distance between the lines

x + 1 = 2 y = - 12 z

and

x = y + 2 = 6 z - 6

(A)2✓ Correct
(B)3
(C) $\frac{5}{2}$
(D) $\frac{3}{2}$

Answer: (A)

Solution

The line

x + 1 = 2 y = - 12 z

passes through

(- 1, 0, 0)

with direction ratios

(12, 6, - 1)

, obtained by clearing fractions, and

x = y + 2 = 6 z - 6

passes through

(0, - 2, 1)

with direction

(6, 6, 1)

. Their cross product is

(12, - 18, 36)

, of magnitude

42

, and the join of the base points is

(1, - 2, 1)

. The shortest distance is

\frac{∣ ( 1 , - 2 , 1 ) \cdot ( 12 , - 18 , 36 ) ∣}{42} = \frac{∣12 + 36 + 36∣}{42} = \frac{84}{42} = 2

JEE Main · 2023 · 25 Jan 2023 · Shift 2 · Single correct option

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