JEE Main · 2023 · 29 Jan 2023 · Shift 2 · Three Dimensional Geometry

Shortest distance between the lines

\frac{x - 1}{2} = \frac{y + 8}{- 7} = \frac{z - 4}{5}

and

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

(A) $23$
(B) $43$ ✓ Correct
(C) $33$
(D) $53$

Answer: (B)

Solution

The lines pass through

a = \hat{i} - 8 \hat{j} + 4 \hat{k}

and

b = \hat{i} + 2 \hat{j} + 6 \hat{k}

with directions

p = 2 \hat{i} - 7 \hat{j} + 5 \hat{k}

and

q = 2 \hat{i} + \hat{j} - 3 \hat{k}

. Their cross product is

p \times q = 16 (\hat{i} + \hat{j} + \hat{k})

, of magnitude

163

. Since

a - b = - 10 \hat{j} - 2 \hat{k}

, the shortest distance is

\frac{( a - b ) \cdot ( p \times q )}{∣ p \times q ∣} = \frac{16 ( 0 - 10 - 2 )}{16 3} = \frac{12}{3} = 43

JEE Main · 2023 · 29 Jan 2023 · Shift 2 · Single correct option

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