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JEE Main 2017 · Mathematics · Three Dimensional Geometry · hard

The shortest distance between the lines (x-1)/(1)=(y-2)/(2)=(z-3)/(2) and (x-2)/(2)=(y-3)/(1)=(z-5)/(-2) is

Single correct option

The shortest distance between the lines x−11=y−22=z−32\dfrac{x-1}{1}=\dfrac{y-2}{2}=\dfrac{z-3}{2} and x−22=y−31=z−5−2\dfrac{x-2}{2}=\dfrac{y-3}{1}=\dfrac{z-5}{-2} is
  1. A. 13\dfrac13
  2. B. 23\dfrac23
  3. C. 11
  4. D. 43\dfrac43
Show answer and solution

Answer: (B) 23\dfrac23

Direction vectors: d⃗1=(1,2,2)\vec d_1=(1,2,2) and d⃗2=(2,1,−2)\vec d_2=(2,1,-2); they are not parallel, so the lines are skew or intersecting. Their cross product is d⃗1×d⃗2=(−6,6,−3)\vec d_1\times\vec d_2=(-6,6,-3), of magnitude 99. The vector joining points on the lines is (2,3,5)−(1,2,3)=(1,1,2)(2,3,5)-(1,2,3)=(1,1,2). Distance =∣(1,1,2)⋅(−6,6,−3)∣9=∣−6+6−6∣9=69=23=\dfrac{|(1,1,2)\cdot(-6,6,-3)|}{9}=\dfrac{|-6+6-6|}{9}=\dfrac69=\dfrac23.
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