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The lines whose direction cosines are given by the relations a l + b m + c n = 0 and m n + n l + l m = 0 are

Options

  1. Aperpendicular if 1 a + 1 b + 1 c = 0
  2. Bperpendicular if a + b + c = 0
  3. Cparallel if 1 a + 1 b + 1 c = 0
  4. Dparallel if a + b + c = 0

Correct answer

A. perpendicular if 1 a + 1 b + 1 c = 0

Step-by-step solution

Given direction cosines are a l + b m + c n = 0     . . . 1 ⇒ n = - a l + b m c     . . . 2 and m n + n l + l m = 0     . . . 3 From equation 1 , - m a l + b m c - l a l + b m c + l m = 0 ⇒ m ( a l + b m ) + l ( a l + b m ) - l c m = 0 ⇒ a l 2 + l m a + b - c + b m 2 = 0 Dividing by m 2 , ⇒ a l 2 m 2 + l m a + b - c + b = 0 If l 1 ,   m 1 ,   n 1 and l 2 ,   m 2 ,   n 2 are direction cosines of two lines, then l 1 m 1 ,   l 1 m 2 are roo

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