Q: The equations of the lines of shortest distance between the lines $ \large \frac{x}{2} = \frac{y}{-3} = \frac{z}{1} $ and $\large \frac{x-2}{3} = \frac{y-1}{-5} = \frac{z+2}{2} $ are

(A) 3(x – 21) = 3y + 92 = 3z – 32

(B) $ \large \frac{x-62/3}{1/3} = \frac{y+31}{1/3} = \frac{z-31}{1/3} $

(C) $ \large \frac{x-21}{1/3} = \frac{y + 92/3}{1/3} = \frac{z-32/3}{1/3} $

(D) $ \large \frac{x-2}{1/3} = \frac{y+3}{1/3} = \frac{z-1}{1/3} $

Sol. Let P(2r_{1} , – 3r_{1}, r_{1}) and Q(3r_{2} + 2, – 5r_{2} + 1, 2r_{2} – 2) be the points on the given lines so that PQ is the line of shortest distance

d.r.s of PQ 2r_{1} – 3r_{2} – 2 , – 3r_{1} + 5r_{2} – 1, r_{1} – 2r_{2} + 2

Since it is perpendicular to given lines

2(2r_{1} – 3r_{2} – 2) – 3(3r_{1} + 5r_{2} – 1) + (r_{1} – 2r_{2} + 2) = 0

and (2r_{1} – 3r_{2} – 2) – 5(– 3r_{1} + 5r_{2} – 1) + 2(r_{1} – 2r_{2} + 2) = 0

r_{1} = 31/3, r_{2} = 19/3

P is (62/3 , -31 , 31/3) and Q is (21 , -92/3 , 32/3)

d.r.s of PQ is $ (\frac{1}{3} , \frac{1}{3} , \frac{1}{3})$

Hence (A), (B) and (C) are the correct answers.