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Maths · Equation of a line; Skew lines, the shortest distance between them and its equation

A line d.c's proportional to (2,1,2) meets each of the lines and Then the coordi

A line d.c's proportional to (2,1,2) meets each of the lines \( \boldsymbol{x}=\boldsymbol{y}+\boldsymbol{a}=\boldsymbol{z} \) and \( x+a=2 y=2 z . \) Then the coordinates of each of the points of intersection are given by

  • A. \( (3 a, 2 a, 3 a) ;(a, a, 2 a) \)
  • B. \( (3 a, 2 a, 3 a) ;(a, a, a) \)
  • C. \( (3 a, 3 a, 3 a) ;(a, a, a) \)
  • D. \( (2 a, 3 a, 3 a) ;(2 a, a, a) \)

Step-by-step solution

The line L1: x = y + a = z, parameterized as (t, t-a, t). L2: x + a = 2y = 2z, parameterized as (s-a, s/2, s/2). A line with direction (2,1,2) through a point on L1: (t+2λ, t-a+λ, t+2λ). For intersection with L2, equate coordinates: t+2λ = s-a, t-a+λ = s/2, t+2λ = s/2. From first and third, s=2a. Then t+2λ = a and t+λ = 2a, giving λ = -a and t = 3a. So intersection points: P = (3a, 2a, 3a) and Q = (a, a, a).
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